---
OA_place: publisher
OA_type: diamond
PlanS_conform: '1'
_id: '20867'
abstract:
- lang: eng
text: We discuss the embeddability of subspaces of the Gromov–Hausdorff space, which
consists of isometry classes of compact metric spaces endowed with the Gromov–Hausdorff
distance, into Hilbert spaces. These embeddings are particularly valuable for
applications to topological data analysis. We prove that its subspace consisting
of metric spaces with at most n points has asymptotic dimension n(n−1)∕2. Thus,
there exists a coarse embedding of that space into a Hilbert space. On the contrary,
if the number of points is not bounded, then the subspace cannot be coarsely embedded
into any uniformly convex Banach space and so, in particular, into any Hilbert
space. Furthermore, we prove that, even if we restrict to finite metric spaces
whose diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz
embedded into any finite-dimensional Hilbert space. We obtain both nonembeddability
results by finding obstructions to coarse and bi-Lipschitz embeddings in families
of isometry classes of finite subsets of the real line endowed with the Euclidean–Hausdorff
distance.
acknowledgement: "The author was supported by the FWF Grant, Project number I4245-N35.
The author would like to thank Thomas Weighill for the helpful discussions around
Theorem 3.10, and Takamitsu Yamauchi for bringing to my attention the fundamental
reference [35]. Furthermore, the author\r\nis thankful for the detailed and helpful
comments of the reviewer of this manuscript."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nicolò
full_name: Zava, Nicolò
id: c8b3499c-7a77-11eb-b046-aa368cbbf2ad
last_name: Zava
orcid: 0000-0001-8686-1888
citation:
ama: Zava N. Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff
space into Hilbert spaces. Algebraic & Geometric Topology. 2025;25(8):5153-5174.
doi:10.2140/agt.2025.25.5153
apa: Zava, N. (2025). Coarse and bi-Lipschitz embeddability of subspaces of the
Gromov–Hausdorff space into Hilbert spaces. Algebraic & Geometric Topology.
Mathematical Sciences Publishers. https://doi.org/10.2140/agt.2025.25.5153
chicago: Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the
Gromov–Hausdorff Space into Hilbert Spaces.” Algebraic & Geometric Topology.
Mathematical Sciences Publishers, 2025. https://doi.org/10.2140/agt.2025.25.5153.
ieee: N. Zava, “Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff
space into Hilbert spaces,” Algebraic & Geometric Topology, vol. 25,
no. 8. Mathematical Sciences Publishers, pp. 5153–5174, 2025.
ista: Zava N. 2025. Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff
space into Hilbert spaces. Algebraic & Geometric Topology. 25(8), 5153–5174.
mla: Zava, Nicolò. “Coarse and Bi-Lipschitz Embeddability of Subspaces of the Gromov–Hausdorff
Space into Hilbert Spaces.” Algebraic & Geometric Topology, vol. 25,
no. 8, Mathematical Sciences Publishers, 2025, pp. 5153–74, doi:10.2140/agt.2025.25.5153.
short: N. Zava, Algebraic & Geometric Topology 25 (2025) 5153–5174.
corr_author: '1'
date_created: 2025-12-29T12:09:09Z
date_published: 2025-11-20T00:00:00Z
date_updated: 2026-01-05T12:19:09Z
day: '20'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.2140/agt.2025.25.5153
external_id:
arxiv:
- '2303.04730'
file:
- access_level: open_access
checksum: 1e05b4f17a44500ae1ae1e21bc636f6a
content_type: application/pdf
creator: dernst
date_created: 2026-01-05T12:16:38Z
date_updated: 2026-01-05T12:16:38Z
file_id: '20943'
file_name: 2025_AlgebraicGeomTopology_Zava.pdf
file_size: 574389
relation: main_file
success: 1
file_date_updated: 2026-01-05T12:16:38Z
has_accepted_license: '1'
intvolume: ' 25'
issue: '8'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '11'
oa: 1
oa_version: Published Version
page: 5153-5174
project:
- _id: 26AD5D90-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I04245
name: Algebraic Footprints of Geometric Features in Homology
publication: Algebraic & Geometric Topology
publication_identifier:
eissn:
- 1472-2739
issn:
- 1472-2747
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff
space into Hilbert spaces
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2025'
...
---
OA_place: repository
_id: '21016'
abstract:
- lang: eng
text: Motivated by applications in chemistry, we give a homlogical definition of
tunnels, or more generally cobordisms, connecting disjoint parts of a cell complex.
For a filtered complex, this defines a persistence module. We give a method for
identifying birth and death times using kernel persistence and a matrix reduction
algorithm for pairing birth and death times.
acknowledgement: "Y. B. B. and L. F. were funded by the Independent Research Fund
Denmark, grant\r\nnumber 1026-00037. T. H. was partially supported by the European
Research Council\r\n(ERC) Horizon 2020, grant number 788183."
article_number: '2505.17858'
article_processing_charge: No
arxiv: 1
author:
- first_name: Yossi
full_name: Bleile, Yossi
id: 920a7385-7995-11ef-9bfd-8c434cd8f3c2
last_name: Bleile
orcid: 0000-0002-4861-9174
- first_name: Lisbeth
full_name: Fajstrup, Lisbeth
last_name: Fajstrup
- first_name: Teresa
full_name: Heiss, Teresa
id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
last_name: Heiss
orcid: 0000-0002-1780-2689
- first_name: Anne Marie
full_name: Svane, Anne Marie
last_name: Svane
- first_name: Søren Strandskov
full_name: Sørensen, Søren Strandskov
last_name: Sørensen
citation:
ama: Bokor Bleile Y, Fajstrup L, Heiss T, Svane AM, Sørensen SS. Identifying cobordisms
using kernel persistence. arXiv. doi:10.48550/arXiv.2505.17858
apa: Bokor Bleile, Y., Fajstrup, L., Heiss, T., Svane, A. M., & Sørensen, S.
S. (n.d.). Identifying cobordisms using kernel persistence. arXiv. https://doi.org/10.48550/arXiv.2505.17858
chicago: Bokor Bleile, Yossi, Lisbeth Fajstrup, Teresa Heiss, Anne Marie Svane,
and Søren Strandskov Sørensen. “Identifying Cobordisms Using Kernel Persistence.”
ArXiv, n.d. https://doi.org/10.48550/arXiv.2505.17858.
ieee: Y. Bokor Bleile, L. Fajstrup, T. Heiss, A. M. Svane, and S. S. Sørensen, “Identifying
cobordisms using kernel persistence,” arXiv. .
ista: Bokor Bleile Y, Fajstrup L, Heiss T, Svane AM, Sørensen SS. Identifying cobordisms
using kernel persistence. arXiv, 2505.17858.
mla: Bokor Bleile, Yossi, et al. “Identifying Cobordisms Using Kernel Persistence.”
ArXiv, 2505.17858, doi:10.48550/arXiv.2505.17858.
short: Y. Bokor Bleile, L. Fajstrup, T. Heiss, A.M. Svane, S.S. Sørensen, ArXiv
(n.d.).
date_created: 2026-01-20T10:12:21Z
date_published: 2025-05-23T00:00:00Z
date_updated: 2026-06-11T11:51:13Z
day: '23'
department:
- _id: HeEd
doi: 10.48550/arXiv.2505.17858
ec_funded: 1
external_id:
arxiv:
- '2505.17858'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2505.17858
month: '05'
oa: 1
oa_version: Preprint
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
publication: arXiv
publication_status: submitted
status: public
title: Identifying cobordisms using kernel persistence
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2025'
...
---
OA_place: repository
_id: '21050'
abstract:
- lang: eng
text: "In 1873, James C. Maxwell conjectured that the electric field generated by
$n$ point charges in generic position has at most $(n-1)^2$ isolated zeroes. The
first (non-optimal) upper bound was only obtained in 2007 by Gabrielov, Novikov
and Shapiro, who also posed two additional interesting conjectures.\r\n In this
article, we give the best upper bound known to date on the number of zeroes of
the electric field, and construct a counterexample to a conjecture of Gabrielov,
Novikov and Shapiro that the number of equilibria cannot exceed those of the distance
function defined by the unit point charges.\r\n Finally, we note that it is quite
possible that Maxwell's quadratic upper bound is not tight, so it is prudent to
find smaller bounds. Hence, we also explore examples and construct configurations
of charges achieving the highest ratios of the number of electric field zeroes
by point charges found to this day."
article_processing_charge: No
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Gonçalo
full_name: Olivera, Gonçalo
last_name: Olivera
citation:
ama: Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic
potential. arXiv. doi:10.48550/ARXIV.2501.05315
apa: Edelsbrunner, H., Fillmore, C. D., & Olivera, G. (n.d.). Counting equilibria
of the electrostatic potential. arXiv. https://doi.org/10.48550/ARXIV.2501.05315
chicago: Edelsbrunner, Herbert, Christopher D Fillmore, and Gonçalo Olivera. “Counting
Equilibria of the Electrostatic Potential.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2501.05315.
ieee: H. Edelsbrunner, C. D. Fillmore, and G. Olivera, “Counting equilibria of the
electrostatic potential,” arXiv. .
ista: Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic
potential. arXiv, 10.48550/ARXIV.2501.05315.
mla: Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.”
ArXiv, doi:10.48550/ARXIV.2501.05315.
short: H. Edelsbrunner, C.D. Fillmore, G. Olivera, ArXiv (n.d.).
corr_author: '1'
date_created: 2026-01-27T14:29:27Z
date_published: 2025-03-20T00:00:00Z
date_updated: 2026-06-02T09:24:17Z
day: '20'
department:
- _id: HeEd
doi: 10.48550/ARXIV.2501.05315
external_id:
arxiv:
- '2501.05315'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2501.05315
month: '03'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
record:
- id: '21021'
relation: dissertation_contains
status: public
- id: '21931'
relation: later_version
status: public
status: public
title: Counting equilibria of the electrostatic potential
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '21253'
abstract:
- lang: eng
text: We solve a problem of Dujmović and Wood (2007) by showing that a complete
convex geometric graph on n vertices cannot be decomposed into fewer than n -
1 star-forests, each consisting of noncrossing edges. This bound is clearly tight.
We also discuss similar questions for abstract graphs.
acknowledgement: A preliminary version of this note has been published in the proceedings
of the 31st International Symposium on Graph Drawing and Network Visualization,
Palermo, 2023. The authors would like to thank the anonymous referees for their
valuable comments.
article_number: '102186'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: János
full_name: Pach, János
last_name: Pach
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
- first_name: Patrick
full_name: Schnider, Patrick
last_name: Schnider
citation:
ama: Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests.
Computational Geometry. 2025;129. doi:10.1016/j.comgeo.2025.102186
apa: Pach, J., Saghafian, M., & Schnider, P. (2025). Decomposition of geometric
graphs into star-forests. Computational Geometry. Elsevier. https://doi.org/10.1016/j.comgeo.2025.102186
chicago: Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of
Geometric Graphs into Star-Forests.” Computational Geometry. Elsevier,
2025. https://doi.org/10.1016/j.comgeo.2025.102186.
ieee: J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs
into star-forests,” Computational Geometry, vol. 129. Elsevier, 2025.
ista: Pach J, Saghafian M, Schnider P. 2025. Decomposition of geometric graphs into
star-forests. Computational Geometry. 129, 102186.
mla: Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.”
Computational Geometry, vol. 129, 102186, Elsevier, 2025, doi:10.1016/j.comgeo.2025.102186.
short: J. Pach, M. Saghafian, P. Schnider, Computational Geometry 129 (2025).
corr_author: '1'
date_created: 2026-02-16T15:48:42Z
date_published: 2025-12-01T00:00:00Z
date_updated: 2026-04-16T09:12:36Z
day: '01'
department:
- _id: HeEd
doi: 10.1016/j.comgeo.2025.102186
external_id:
arxiv:
- '2306.13201'
intvolume: ' 129'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2306.13201
month: '12'
oa: 1
oa_version: Preprint
publication: Computational Geometry
publication_identifier:
issn:
- 0925-7721
publication_status: published
publisher: Elsevier
quality_controlled: '1'
related_material:
record:
- id: '15012'
relation: earlier_version
status: public
status: public
title: Decomposition of geometric graphs into star-forests
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 129
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
_id: '17149'
abstract:
- lang: eng
text: The approximation of a circle with the edges of a fine square grid distorts
the perimeter by a factor about 4/Pi. We prove that this factor is the same on
average (in the ergodic sense) for approximations of any rectifiable curve by
the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend
the results to all dimensions, generalizing Voronoi paths to Voronoi scapes.
acknowledgement: "The authors thank Ranita Biswas and Tatiana Ezubova for the collaboration
on computational experiments that motivated the work reported in this paper. The
authors also thank Daniel Bonnema for proofreading and noticing an issue with the
original proof of Lemma 4.3.\r\nOpen access funding provided by Institute of Science
and Technology (IST Austria).\r\nThis project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund
(FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109,
‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No.
I 02979-N35."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Anton
full_name: Nikitenko, Anton
id: 3E4FF1BA-F248-11E8-B48F-1D18A9856A87
last_name: Nikitenko
orcid: 0000-0002-0659-3201
citation:
ama: Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths
and scapes. Discrete & Computational Geometry. 2025;73:490-499. doi:10.1007/s00454-024-00660-y
apa: Edelsbrunner, H., & Nikitenko, A. (2025). Average and expected distortion
of Voronoi paths and scapes. Discrete & Computational Geometry. Springer
Nature. https://doi.org/10.1007/s00454-024-00660-y
chicago: Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion
of Voronoi Paths and Scapes.” Discrete & Computational Geometry. Springer
Nature, 2025. https://doi.org/10.1007/s00454-024-00660-y.
ieee: H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi
paths and scapes,” Discrete & Computational Geometry, vol. 73. Springer
Nature, pp. 490–499, 2025.
ista: Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi
paths and scapes. Discrete & Computational Geometry. 73, 490–499.
mla: Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion
of Voronoi Paths and Scapes.” Discrete & Computational Geometry, vol.
73, Springer Nature, 2025, pp. 490–99, doi:10.1007/s00454-024-00660-y.
short: H. Edelsbrunner, A. Nikitenko, Discrete & Computational Geometry 73 (2025)
490–499.
corr_author: '1'
date_created: 2024-06-16T22:01:07Z
date_published: 2025-03-01T00:00:00Z
date_updated: 2026-02-16T12:18:50Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-024-00660-y
ec_funded: 1
external_id:
arxiv:
- '2012.03350'
isi:
- '001238566200004'
pmid:
- '39974750'
file:
- access_level: open_access
checksum: ffb0c818222138f9f113f4bbea41e834
content_type: application/pdf
creator: dernst
date_created: 2025-04-23T07:31:32Z
date_updated: 2025-04-23T07:31:32Z
file_id: '19610'
file_name: 2025_DiscreteComputGeom_EdelsbrunnerHe.pdf
file_size: 283443
relation: main_file
success: 1
file_date_updated: 2025-04-23T07:31:32Z
has_accepted_license: '1'
intvolume: ' 73'
isi: 1
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 490-499
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Discrete & Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Average and expected distortion of Voronoi paths and scapes
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 73
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '18626'
abstract:
- lang: eng
text: "The local angle property of the (order-1) Delaunay triangulations of a generic
set in R2\r\n asserts that the sum of two angles opposite a common edge is less
than π. This paper extends this property to higher order and uses it to generalize
two classic properties from order-1 to order-2: (1) among the complete level-2
hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation
lexicographically maximizes the sorted angle vector; (2) among the maximal level-2
hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation
is the only one that has the local angle property. We also use our method of establishing
(2) to give a new short proof of the angle vector optimality for the (order-1)
Delaunay triangulation. For order-1, both properties have been instrumental in
numerous applications of Delaunay triangulations, and we expect that their generalization
will make order-2 Delaunay triangulations more attractive to applications as well."
acknowledgement: Work by the first and third authors is partially supported by the
European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian
Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center
TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second
author is partially supported by the Alexander von Humboldt Foundation.
article_number: '110055'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Alexey
full_name: Garber, Alexey
last_name: Garber
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize
angles. Advances in Mathematics. 2025;461. doi:10.1016/j.aim.2024.110055
apa: Edelsbrunner, H., Garber, A., & Saghafian, M. (2025). Order-2 Delaunay
triangulations optimize angles. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2024.110055
chicago: Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay
Triangulations Optimize Angles.” Advances in Mathematics. Elsevier, 2025.
https://doi.org/10.1016/j.aim.2024.110055.
ieee: H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations
optimize angles,” Advances in Mathematics, vol. 461. Elsevier, 2025.
ista: Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations
optimize angles. Advances in Mathematics. 461, 110055.
mla: Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.”
Advances in Mathematics, vol. 461, 110055, Elsevier, 2025, doi:10.1016/j.aim.2024.110055.
short: H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025).
corr_author: '1'
date_created: 2024-12-08T23:01:54Z
date_published: 2025-02-01T00:00:00Z
date_updated: 2025-04-15T07:16:53Z
day: '01'
department:
- _id: HeEd
doi: 10.1016/j.aim.2024.110055
ec_funded: 1
external_id:
arxiv:
- '2310.18238'
isi:
- '001370682500001'
intvolume: ' 461'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2310.18238
month: '02'
oa: 1
oa_version: Preprint
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Advances in Mathematics
publication_identifier:
eissn:
- 1090-2082
issn:
- 0001-8708
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Order-2 Delaunay triangulations optimize angles
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 461
year: '2025'
...
---
OA_place: publisher
_id: '18979'
abstract:
- lang: eng
text: "Topological Data Analysis (TDA) is a discipline utilizing the mathematical
field of topology to study data, most prominently collections of point sets. This
thesis summarizes three projects related to computations in TDA.\r\n\r\nThe first
one establishes a variant of TDA for chromatic point sets, where each point is
given a color. For example, we are given positions of cells within a tumor microenvironment,
and color the cancerous cells red, and the immune cells blue.\r\n\r\nThe aim is
then to give a quantitative description of how the two or more sets of points
spatially interact. Building on image, kernel and cokernel variants of persistent
homology, we suggest six-packs of persistent diagrams as such a descriptor.\r\n\r\nWe
describe a construction of a chromatic alpha complex, which enables efficient
computation of several variants of the six-packs. We give topological descriptions
of natural subcomplexes of the chromatic alpha complex, and show that the radii
of the simplices form a discrete Morse function. Finally, we provide an implementation
of the presented chromatic TDA pipeline.\r\n\r\nThe second part aims to translate
a powerful tool of sheaf theory to elementary terms using labeled matrices. The
goal is to enable their use in computational settings. We show that derived categories
of sheaves over finite posets have, up to isomorphism, unique objects---minimal
injective resolutions---and give a concrete algorithm to compute them. We further
describe simple algorithms to compute derived pushforwards and pullbacks for monotonic
maps, and their proper variants for inclusions, and demonstrate their tractability
by providing an implementation. Finally, we suggest a discrete definition of microsupport
and show desirable properties inspired by discrete Morse theory.\r\n\r\nIn the
last part, we present a collection of observations about collapses. We give a
characterization of collapsibility in terms of unitriangular submatrices of the
boundary matrix, a cotree-tree decomposition, and the optimal solution to a variant
of the Procrustes problem. We establish relation between dual collapses and relative
Morse theory and pose several open questions. Finally, focusing on complexes embedded
in the three-dimensional Euclidean space, we describe a relation between the collapsibility
and the triviality of a polygonal knot."
acknowledgement: "The research presented in this thesis was funded with the Wittgenstein
Prize,\r\nAustrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative
Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian
Science Fund (FWF),\r\ngrant no. I 02979-N35.\r\n"
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Ondrej
full_name: Draganov, Ondrej
id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
last_name: Draganov
orcid: 0000-0003-0464-3823
citation:
ama: Draganov O. Structures and computations in topological data analysis. 2025.
doi:10.15479/at:ista:18979
apa: Draganov, O. (2025). Structures and computations in topological data analysis.
Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:18979
chicago: Draganov, Ondrej. “Structures and Computations in Topological Data Analysis.”
Institute of Science and Technology Austria, 2025. https://doi.org/10.15479/at:ista:18979.
ieee: O. Draganov, “Structures and computations in topological data analysis,” Institute
of Science and Technology Austria, 2025.
ista: Draganov O. 2025. Structures and computations in topological data analysis.
Institute of Science and Technology Austria.
mla: Draganov, Ondrej. Structures and Computations in Topological Data Analysis.
Institute of Science and Technology Austria, 2025, doi:10.15479/at:ista:18979.
short: O. Draganov, Structures and Computations in Topological Data Analysis, Institute
of Science and Technology Austria, 2025.
corr_author: '1'
date_created: 2025-01-31T17:04:40Z
date_published: 2025-02-03T00:00:00Z
date_updated: 2026-04-07T11:47:30Z
day: '03'
ddc:
- '514'
- '004'
degree_awarded: PhD
department:
- _id: GradSch
- _id: HeEd
doi: 10.15479/at:ista:18979
file:
- access_level: closed
checksum: af6567e5d35e5eb330b8925ae37f1998
content_type: application/zip
creator: odragano
date_created: 2025-01-31T16:58:30Z
date_updated: 2025-01-31T16:58:30Z
file_id: '18983'
file_name: Thesis.zip
file_size: 11899491
relation: source_file
- access_level: open_access
checksum: c3fef68e35b9dc2020b2ca6006da6343
content_type: application/pdf
creator: odragano
date_created: 2025-02-04T16:22:07Z
date_updated: 2025-02-04T16:22:07Z
file_id: '19000'
file_name: Thesis.pdf
file_size: 8857514
relation: main_file
file_date_updated: 2025-02-04T16:22:07Z
has_accepted_license: '1'
keyword:
- topological data analysis
- chromatic point set
- alpha complex
- persistent homology
- six pack
- sheaf
- microlocal discrete Morse
- injective resolution
- collapse
- knot
- discrete Morse theory
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: '140'
project:
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
record:
- id: '15091'
relation: part_of_dissertation
status: public
- id: '18981'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
title: Structures and computations in topological data analysis
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2025'
...
---
OA_type: closed access
_id: '19937'
abstract:
- lang: eng
text: Simplets are elementary units within simplicial complexes and are fundamental
for analyzing the structure of simplicial complexes. Previous efforts have mainly
focused on accurately counting or approximating the number of simplets rather
than studying their frequencies. However, analyzing simplet frequencies is more
practical for large-scale simplicial complexes. This paper introduces the Simplet
Frequency Distribution (SFD) vector, which enables the analysis of simplet frequencies
in simplicial complexes. Additionally, we provide a bound on the sample complexity
required to approximate the SFD vector using any uniform sampling-based algorithm
accurately. We extend the definition of simplet frequency distribution to encompass
simplices, allowing for the analysis of simplet frequencies within simplices of
simplicial complexes. This paper introduces the Simplet Degree Vector (SDV) and
the Simplet Degree Centrality (SDC), facilitating this analysis for each simplex.
Furthermore, we present a bound on the sample complexity required for accurately
approximating the SDV and SDC for a set of simplices using any uniform sampling-based
algorithm. We also introduce algorithms for approximating SFD, geometric SFD,
SDV, and SDC. We also validate the theoretical bounds with experiments on random
simplicial complexes and demonstrate the practical application through a case
study.
acknowledgement: "The authors would like to thank the anonymous reviewers for their
valuable comments and suggestions, which improved this paper.\r\nWork by the first
and fourth authors is partially supported by the European Research Council (ERC),
grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant
no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science
Fund (FWF), grant no. I 02979-N35."
article_number: '122425'
article_processing_charge: No
article_type: original
author:
- first_name: Mohammad
full_name: Mahini, Mohammad
last_name: Mahini
- first_name: Hamid
full_name: Beigy, Hamid
last_name: Beigy
- first_name: Salman
full_name: Qadami, Salman
last_name: Qadami
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: 'Mahini M, Beigy H, Qadami S, Saghafian M. Simplet-based signatures and approximation
in simplicial complexes: Frequency, degree, and centrality. Information Sciences.
2025;719(11). doi:10.1016/j.ins.2025.122425'
apa: 'Mahini, M., Beigy, H., Qadami, S., & Saghafian, M. (2025). Simplet-based
signatures and approximation in simplicial complexes: Frequency, degree, and centrality.
Information Sciences. Elsevier. https://doi.org/10.1016/j.ins.2025.122425'
chicago: 'Mahini, Mohammad, Hamid Beigy, Salman Qadami, and Morteza Saghafian. “Simplet-Based
Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.”
Information Sciences. Elsevier, 2025. https://doi.org/10.1016/j.ins.2025.122425.'
ieee: 'M. Mahini, H. Beigy, S. Qadami, and M. Saghafian, “Simplet-based signatures
and approximation in simplicial complexes: Frequency, degree, and centrality,”
Information Sciences, vol. 719, no. 11. Elsevier, 2025.'
ista: 'Mahini M, Beigy H, Qadami S, Saghafian M. 2025. Simplet-based signatures
and approximation in simplicial complexes: Frequency, degree, and centrality.
Information Sciences. 719(11), 122425.'
mla: 'Mahini, Mohammad, et al. “Simplet-Based Signatures and Approximation in Simplicial
Complexes: Frequency, Degree, and Centrality.” Information Sciences, vol.
719, no. 11, 122425, Elsevier, 2025, doi:10.1016/j.ins.2025.122425.'
short: M. Mahini, H. Beigy, S. Qadami, M. Saghafian, Information Sciences 719 (2025).
corr_author: '1'
date_created: 2025-06-30T08:48:48Z
date_published: 2025-11-01T00:00:00Z
date_updated: 2025-12-30T09:05:32Z
day: '01'
department:
- _id: HeEd
doi: 10.1016/j.ins.2025.122425
ec_funded: 1
external_id:
isi:
- '001516170500002'
intvolume: ' 719'
isi: 1
issue: '11'
language:
- iso: eng
month: '11'
oa_version: None
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Information Sciences
publication_identifier:
issn:
- 0020-0255
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Simplet-based signatures and approximation in simplicial complexes: Frequency,
degree, and centrality'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 719
year: '2025'
...
---
OA_place: publisher
OA_type: gold
_id: '20005'
abstract:
- lang: eng
text: "We generalize a classical result by Boris Delaunay that introduced Delaunay
triangulations. In particular, we prove that for a locally finite and coarsely
dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly
binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose
exactly k points of A. We extend this result to the cases in which the points
are weighted, and when A contains only finitely many points in ℝ^d or in \U0001D54A^d.
Furthermore, we use the result to give a new geometric proof for the fact that
volumes of hypersimplices are Eulerian numbers."
acknowledgement: "Herbert Edelsbrunner: partially supported by the Wittgenstein Prize,
Austrian Science\r\nFund (FWF), grant no. Z 342-N31, and by the DFG Collaborative
Research Center TRR 109,\r\nAustrian Science Fund (FWF), grant no. I 02979-N35.\r\nAlexey
Garber: partially supported by the Simons Foundation.\r\nMorteza Saghafian: partially
supported by the Wittgenstein Prize, Austrian Science Fund (FWF),\r\ngrant no. Z
342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund
(FWF), grant no. I 02979-N35"
alternative_title:
- LIPIcs
article_number: '43'
article_processing_charge: Yes
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Alexey
full_name: Garber, Alexey
last_name: Garber
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: 'Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In:
41st International Symposium on Computational Geometry. Vol 332. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:10.4230/LIPIcs.SoCG.2025.43'
apa: 'Edelsbrunner, H., Garber, A., & Saghafian, M. (2025). On spheres with
k points inside. In 41st International Symposium on Computational Geometry
(Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
https://doi.org/10.4230/LIPIcs.SoCG.2025.43'
chicago: Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres
with k Points Inside.” In 41st International Symposium on Computational Geometry,
Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. https://doi.org/10.4230/LIPIcs.SoCG.2025.43.
ieee: H. Edelsbrunner, A. Garber, and M. Saghafian, “On spheres with k points inside,”
in 41st International Symposium on Computational Geometry, Kanazawa, Japan,
2025, vol. 332.
ista: 'Edelsbrunner H, Garber A, Saghafian M. 2025. On spheres with k points inside.
41st International Symposium on Computational Geometry. SoCG: Symposium on Computational
Geometry, LIPIcs, vol. 332, 43.'
mla: Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” 41st International
Symposium on Computational Geometry, vol. 332, 43, Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2025, doi:10.4230/LIPIcs.SoCG.2025.43.
short: H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium
on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2025.
conference:
end_date: 2025-06-27
location: Kanazawa, Japan
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2025-06-23
corr_author: '1'
date_created: 2025-07-13T22:01:22Z
date_published: 2025-06-20T00:00:00Z
date_updated: 2025-07-14T07:26:14Z
day: '20'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2025.43
external_id:
arxiv:
- '2410.21204'
file:
- access_level: open_access
checksum: b5313ed8575ea87913c71a6e3c7513c8
content_type: application/pdf
creator: dernst
date_created: 2025-07-14T07:24:22Z
date_updated: 2025-07-14T07:24:22Z
file_id: '20016'
file_name: 2025_LIPIcs.SoCG_Edelsbrunner.pdf
file_size: 661893
relation: main_file
success: 1
file_date_updated: 2025-07-14T07:24:22Z
has_accepted_license: '1'
intvolume: ' 332'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: 41st International Symposium on Computational Geometry
publication_identifier:
eissn:
- 1868-8969
isbn:
- '9783959773706'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: On spheres with k points inside
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 332
year: '2025'
...
---
OA_place: publisher
OA_type: gold
_id: '20006'
abstract:
- lang: eng
text: In numerous fields, dynamic time series data require continuous updates, necessitating
efficient data processing techniques for accurate analysis. This paper examines
the banana tree data structure, specifically designed to efficiently maintain
the multi-scale topological descriptor commonly known as persistent homology for
dynamically changing time series data. We implement this data structure and conduct
an experimental study to assess its properties and runtime for update operations.
Our findings indicate that banana trees are highly effective with unbiased random
data, outperforming state-of-the-art static algorithms in these scenarios. Additionally,
our results show that real-world time series share structural properties with
unbiased random walks, suggesting potential practical utility for our implementation.
acknowledgement: "Lara Ost: Supported by the Vienna Graduate School on Computational
Optimization\r\n(VGSCO), FWF project no. W1260-N35.\r\nSebastiano Cultrera di Montesano:
Supported by the Eric and Wendy Schmidt Center at the Broad Institute of MIT and
Harvard.\r\nHerbert Edelsbrunner: Partially supported by the Wittgenstein Prize,
FWF grant no. Z 342-N31,\r\nand by the DFG Collaborative Research Center TRR 109,
FWF grant no. I 02979-N35."
alternative_title:
- LIPIcs
article_number: '71'
article_processing_charge: Yes
arxiv: 1
author:
- first_name: Lara
full_name: Ost, Lara
last_name: Ost
- first_name: Sebastiano
full_name: Cultrera di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera di Montesano
orcid: 0000-0001-6249-0832
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
citation:
ama: 'Ost L, Cultrera di Montesano S, Edelsbrunner H. Banana trees for the persistence
in time series experimentally. In: 41st International Symposium on Computational
Geometry. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025.
doi:10.4230/LIPIcs.SoCG.2025.71'
apa: 'Ost, L., Cultrera di Montesano, S., & Edelsbrunner, H. (2025). Banana
trees for the persistence in time series experimentally. In 41st International
Symposium on Computational Geometry (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl
- Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2025.71'
chicago: Ost, Lara, Sebastiano Cultrera di Montesano, and Herbert Edelsbrunner.
“Banana Trees for the Persistence in Time Series Experimentally.” In 41st International
Symposium on Computational Geometry, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2025. https://doi.org/10.4230/LIPIcs.SoCG.2025.71.
ieee: L. Ost, S. Cultrera di Montesano, and H. Edelsbrunner, “Banana trees for the
persistence in time series experimentally,” in 41st International Symposium
on Computational Geometry, Kanazawa, Japan, 2025, vol. 332.
ista: 'Ost L, Cultrera di Montesano S, Edelsbrunner H. 2025. Banana trees for the
persistence in time series experimentally. 41st International Symposium on Computational
Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 71.'
mla: Ost, Lara, et al. “Banana Trees for the Persistence in Time Series Experimentally.”
41st International Symposium on Computational Geometry, vol. 332, 71, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:10.4230/LIPIcs.SoCG.2025.71.
short: L. Ost, S. Cultrera di Montesano, H. Edelsbrunner, in:, 41st International
Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2025.
conference:
end_date: 2025-06-27
location: Kanazawa, Japan
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2025-06-23
corr_author: '1'
date_created: 2025-07-13T22:01:22Z
date_published: 2025-06-20T00:00:00Z
date_updated: 2025-12-30T11:04:33Z
day: '20'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2025.71
external_id:
arxiv:
- '2405.17920'
file:
- access_level: open_access
checksum: 3a4a7a707a56e0cfdf51428782dee55a
content_type: application/pdf
creator: dernst
date_created: 2025-07-14T08:23:38Z
date_updated: 2025-07-14T08:23:38Z
file_id: '20017'
file_name: 2025_LIPIcs.SoCG_Ost.pdf
file_size: 834623
relation: main_file
success: 1
file_date_updated: 2025-07-14T08:23:38Z
has_accepted_license: '1'
intvolume: ' 332'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 9B9290DE-BA93-11EA-9121-9846C619BF3A
grant_number: W1260-N35
name: Vienna Graduate School on Computational Optimization
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
publication: 41st International Symposium on Computational Geometry
publication_identifier:
eissn:
- 1868-8969
isbn:
- '9783959773706'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
related_material:
link:
- relation: software
url: https://github.com/laraost/BananaPersist
scopus_import: '1'
status: public
title: Banana trees for the persistence in time series experimentally
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 332
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20260'
abstract:
- lang: eng
text: The medial axis of a set consists of the points in the ambient space without
a unique closest point in the original set. Since its introduction, the medial
axis has been used extensively in many applications as a method of computing a
skeleton topologically equivalent to the original set. Unfortunately, one limiting
factor in the use of the medial axis of a smooth manifold is that it is not necessarily
topologically stable under small perturbations of the manifold. To counter these
instabilities, various prunings of the medial axis have been proposed in the computational
geometry community. Here, we examine one type of pruning, called burning. Because
of the good experimental results it was hoped that the burning method of simplifying
the medial axis would be stable. In this work, we show a simple example that dashes
such hopes. Based on Bing’s house with two rooms, we demonstrate an isotopy of
a shape where the medial axis goes from collapsible to non-collapsible. More precisely,
we consider the standard deformation retract from the closed ball to Bing’s house
with two rooms, but stop just short of the point where Bing’s house becomes two
dimensional. This way we obtain an isotopy from the 3-ball to a thickened version
of Bing’s house. Under this isotopy, the medial axis goes from collapsible to
non-collapsible. We stress that this isotopy can be made generic, in the sense
of singularity theory, as developed by Arnol’d and Thom.
acknowledgement: "We thank André Lieutier, David Letscher, Ellen Gasparovic, Kathryn
Leonard, and Tao Ju for early discussions on this work. We also thank Lu Liu, Yajie
Yan, and Tao Ju for sharing code to generate the examples. We further thank Abigail
Thompson for discussion on the conjecture and James Damon for sharing his insight
in singularity theory. We thank the reviewers for their detailed reviews, which
helped to improve the exposition.\r\nOpen access funding provided by Institute of
Science and Technology (IST Austria). Partially supported by the DFG Collaborative
Research Center TRR 109, ‘Discretization in Geometry and Dynamics’ and the European
Research Council (ERC), grant no. 788183, ‘Alpha Shape Theory Extended’. The first
author was supported in part by the National Science Foundation through grants DBI-1759807,
CCF-1907612, and CCF-2444309. The fourth author was supported by the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
grant agreement No. 754411, the Austrian science fund (FWF) M-3073, ANR grant StratMesh,
ANR-24-CE48-1899, and the welcome package from IDEX of the Université Côte d’Azur,
ANR-15-IDEX-01."
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Erin Wolf
full_name: Chambers, Erin Wolf
last_name: Chambers
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Burning or collapsing
the medial axis is unstable. La Matematica. 2025;4:811-828. doi:10.1007/s44007-025-00170-0
apa: Chambers, E. W., Fillmore, C. D., Stephenson, E. R., & Wintraecken, M.
(2025). Burning or collapsing the medial axis is unstable. La Matematica.
Springer Nature. https://doi.org/10.1007/s44007-025-00170-0
chicago: Chambers, Erin Wolf, Christopher D Fillmore, Elizabeth R Stephenson, and
Mathijs Wintraecken. “Burning or Collapsing the Medial Axis Is Unstable.” La
Matematica. Springer Nature, 2025. https://doi.org/10.1007/s44007-025-00170-0.
ieee: E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Burning
or collapsing the medial axis is unstable,” La Matematica, vol. 4. Springer
Nature, pp. 811–828, 2025.
ista: Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing
the medial axis is unstable. La Matematica. 4, 811–828.
mla: Chambers, Erin Wolf, et al. “Burning or Collapsing the Medial Axis Is Unstable.”
La Matematica, vol. 4, Springer Nature, 2025, pp. 811–28, doi:10.1007/s44007-025-00170-0.
short: E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, La Matematica
4 (2025) 811–828.
corr_author: '1'
date_created: 2025-08-31T22:01:33Z
date_published: 2025-12-01T00:00:00Z
date_updated: 2026-04-07T11:42:48Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s44007-025-00170-0
ec_funded: 1
file:
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content_type: application/pdf
creator: dernst
date_created: 2025-12-30T07:52:58Z
date_updated: 2025-12-30T07:52:58Z
file_id: '20885'
file_name: 2025_LaMatematica_Chambers.pdf
file_size: 2678640
relation: main_file
success: 1
file_date_updated: 2025-12-30T07:52:58Z
has_accepted_license: '1'
intvolume: ' 4'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 811-828
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
publication: La Matematica
publication_identifier:
eissn:
- 2730-9657
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
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- id: '21021'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Burning or collapsing the medial axis is unstable
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 4
year: '2025'
...
---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
PlanS_conform: '1'
_id: '20293'
abstract:
- lang: eng
text: Motivated by questions arising at the intersection of information theory and
geometry, we compare two dissimilarity measures between finite categorical distributions.
One is the well-known Jensen–Shannon divergence, which is easy to compute and
whose square root is a proper metric. The other is what we call the minmax divergence,
which is harder to compute. Just like the Jensen–Shannon divergence, it arises
naturally from the Kullback–Leibler divergence. The main contribution of this
paper is a proof showing that the minmax divergence can be tightly approximated
by the Jensen–Shannon divergence. The bounds suggest that the square root of the
minmax divergence is a metric, and we prove that this is indeed true in the one-dimensional
case. The general case remains open. Finally, we consider analogous questions
in the context of another Bregman divergence and the corresponding Burbea–Rao
(Jensen–Bregman) divergence.
acknowledgement: "This research received partial funding from the European Research
Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation
programme, grant no. 788183, the\r\nWittgenstein Prize, Austrian Science Fund (FWF),
grant no. Z 342-N31, the DFG Collaborative\r\nResearch Center TRR 109, ‘Discretization
in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35, and
the 2022 Google Research Scholar Award for project ‘Algorithms for Topological Analysis
of Neural Networks’. The APC was waived."
article_number: '854'
article_processing_charge: Yes
article_type: original
author:
- first_name: Arseniy
full_name: Akopyan, Arseniy
id: 430D2C90-F248-11E8-B48F-1D18A9856A87
last_name: Akopyan
orcid: 0000-0002-2548-617X
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Ziga
full_name: Virk, Ziga
id: 2E36B656-F248-11E8-B48F-1D18A9856A87
last_name: Virk
- first_name: Hubert
full_name: Wagner, Hubert
id: 379CA8B8-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
citation:
ama: Akopyan A, Edelsbrunner H, Virk Z, Wagner H. Tight bounds between the Jensen–Shannon
divergence and the minmax divergence. Entropy. 2025;27(8). doi:10.3390/e27080854
apa: Akopyan, A., Edelsbrunner, H., Virk, Z., & Wagner, H. (2025). Tight bounds
between the Jensen–Shannon divergence and the minmax divergence. Entropy.
MDPI. https://doi.org/10.3390/e27080854
chicago: Akopyan, Arseniy, Herbert Edelsbrunner, Ziga Virk, and Hubert Wagner. “Tight
Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” Entropy.
MDPI, 2025. https://doi.org/10.3390/e27080854.
ieee: A. Akopyan, H. Edelsbrunner, Z. Virk, and H. Wagner, “Tight bounds between
the Jensen–Shannon divergence and the minmax divergence,” Entropy, vol.
27, no. 8. MDPI, 2025.
ista: Akopyan A, Edelsbrunner H, Virk Z, Wagner H. 2025. Tight bounds between the
Jensen–Shannon divergence and the minmax divergence. Entropy. 27(8), 854.
mla: Akopyan, Arseniy, et al. “Tight Bounds between the Jensen–Shannon Divergence
and the Minmax Divergence.” Entropy, vol. 27, no. 8, 854, MDPI, 2025, doi:10.3390/e27080854.
short: A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025).
corr_author: '1'
date_created: 2025-09-07T22:01:33Z
date_published: 2025-08-01T00:00:00Z
date_updated: 2025-09-30T14:32:31Z
day: '01'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.3390/e27080854
ec_funded: 1
external_id:
isi:
- '001557476000001'
pmid:
- '40870326'
file:
- access_level: open_access
checksum: 65c5399c4015d9c8abb8c7a96f3d7836
content_type: application/pdf
creator: dernst
date_created: 2025-09-08T07:55:48Z
date_updated: 2025-09-08T07:55:48Z
file_id: '20309'
file_name: 2025_Entropy_Akopyan.pdf
file_size: 379340
relation: main_file
success: 1
file_date_updated: 2025-09-08T07:55:48Z
has_accepted_license: '1'
intvolume: ' 27'
isi: 1
issue: '8'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Entropy
publication_identifier:
eissn:
- 1099-4300
publication_status: published
publisher: MDPI
quality_controlled: '1'
scopus_import: '1'
status: public
title: Tight bounds between the Jensen–Shannon divergence and the minmax divergence
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 27
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20323'
abstract:
- lang: eng
text: We establish several results combining discrete Morse theory and microlocal
sheaf theory in the setting of finite posets and simplicial complexes. Our primary
tool is a computationally tractable description of the bounded derived category
of sheaves on a poset with the Alexandrov topology. We prove that each bounded
complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes)
minimal injective resolution, and we provide algorithms for computing minimal
injective resolution of an injective complex, as well as several useful functors
between derived categories of sheaves. For the constant sheaf on a simplicial
complex, we give asymptotically tight bounds on the complexity of computing the
minimal injective resolution using those algorithms. Our main result is a novel
definition of the discrete microsupport of a bounded complex of sheaves on a finite
poset. We detail several foundational properties of the discrete microsupport,
as well as a microlocal generalization of the discrete homological Morse theorem
and Morse inequalities.
acknowledgement: This project has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation programme,
grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant
no. Z342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization
in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35
article_number: '108068'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Adam
full_name: Brown, Adam
id: 70B7FDF6-608D-11E9-9333-8535E6697425
last_name: Brown
- first_name: Ondrej
full_name: Draganov, Ondrej
id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
last_name: Draganov
orcid: 0000-0003-0464-3823
citation:
ama: Brown A, Draganov O. Discrete microlocal Morse theory. Journal of Pure and
Applied Algebra. 2025;229(10). doi:10.1016/j.jpaa.2025.108068
apa: Brown, A., & Draganov, O. (2025). Discrete microlocal Morse theory. Journal
of Pure and Applied Algebra. Elsevier. https://doi.org/10.1016/j.jpaa.2025.108068
chicago: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” Journal
of Pure and Applied Algebra. Elsevier, 2025. https://doi.org/10.1016/j.jpaa.2025.108068.
ieee: A. Brown and O. Draganov, “Discrete microlocal Morse theory,” Journal of
Pure and Applied Algebra, vol. 229, no. 10. Elsevier, 2025.
ista: Brown A, Draganov O. 2025. Discrete microlocal Morse theory. Journal of Pure
and Applied Algebra. 229(10), 108068.
mla: Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” Journal
of Pure and Applied Algebra, vol. 229, no. 10, 108068, Elsevier, 2025, doi:10.1016/j.jpaa.2025.108068.
short: A. Brown, O. Draganov, Journal of Pure and Applied Algebra 229 (2025).
corr_author: '1'
date_created: 2025-09-10T05:40:09Z
date_published: 2025-10-01T00:00:00Z
date_updated: 2025-12-30T07:55:21Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1016/j.jpaa.2025.108068
ec_funded: 1
external_id:
arxiv:
- '2209.14993'
file:
- access_level: open_access
checksum: 39bcad462278c9322ef810af7db67f56
content_type: application/pdf
creator: dernst
date_created: 2025-12-30T07:55:08Z
date_updated: 2025-12-30T07:55:08Z
file_id: '20886'
file_name: 2025_JourPureAppliedAlgebra_Brown.pdf
file_size: 3090836
relation: main_file
success: 1
file_date_updated: 2025-12-30T07:55:08Z
has_accepted_license: '1'
intvolume: ' 229'
issue: '10'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Journal of Pure and Applied Algebra
publication_identifier:
issn:
- 0022-4049
publication_status: published
publisher: Elsevier
quality_controlled: '1'
related_material:
record:
- id: '18981'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Discrete microlocal Morse theory
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 229
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '20490'
abstract:
- lang: eng
text: "We study flips in hypertriangulations of planar points sets. Here a level-k
hypertriangulation of n\r\n points in the plane is a subdivision induced by the
projection of a k-hypersimplex, which is the convex hull of the barycenters of
the (k-1)-dimensional faces of the standard (n-1)-simplex. In particular, we introduce
four types of flips and prove that the level-2 hypertriangulations are connected
by these flips.\r\n"
acknowledgement: Work by all authors but the second is supported by the European Research
Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund
(FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,
Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is
partially supported by the Alexander von Humboldt Foundation and by the Simons Foundation
. The second author thanks Jesús A. De Loera for useful discussions on flips and
non-flips and Pavel Galashin and Alexey Balitskiy for useful discussions on plabic
graphs.
article_number: '104248'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Alexey
full_name: Garber, Alexey
last_name: Garber
- first_name: Mohadese
full_name: Ghafari, Mohadese
last_name: Ghafari
- first_name: Teresa
full_name: Heiss, Teresa
id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
last_name: Heiss
orcid: 0000-0002-1780-2689
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional
hypertriangulations. European Journal of Combinatorics. 2025;132. doi:10.1016/j.ejc.2025.104248
apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2025).
Flips in two-dimensional hypertriangulations. European Journal of Combinatorics.
Elsevier. https://doi.org/10.1016/j.ejc.2025.104248
chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and
Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” European
Journal of Combinatorics. Elsevier, 2025. https://doi.org/10.1016/j.ejc.2025.104248.
ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “Flips
in two-dimensional hypertriangulations,” European Journal of Combinatorics,
vol. 132. Elsevier, 2025.
ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2025. Flips in
two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248.
mla: Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.”
European Journal of Combinatorics, vol. 132, 104248, Elsevier, 2025, doi:10.1016/j.ejc.2025.104248.
short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European
Journal of Combinatorics 132 (2025).
corr_author: '1'
date_created: 2025-10-19T22:01:31Z
date_published: 2025-10-10T00:00:00Z
date_updated: 2025-12-01T12:57:29Z
day: '10'
department:
- _id: HeEd
doi: 10.1016/j.ejc.2025.104248
ec_funded: 1
external_id:
arxiv:
- '2212.11380'
isi:
- '001599061500002'
intvolume: ' 132'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2212.11380
month: '10'
oa: 1
oa_version: Preprint
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: European Journal of Combinatorics
publication_identifier:
issn:
- 0195-6698
publication_status: epub_ahead
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Flips in two-dimensional hypertriangulations
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 132
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '20585'
abstract:
- lang: eng
text: Motivated by applications in medical sciences, we study finite chromatic sets
in Euclidean space from a topological perspective. Based on the persistent homology
for images, kernels and cokernels, we design provably stable homological quantifiers
that describe the geometric micro- and macro-structure of how the color classes
mingle. These can be efficiently computed using chromatic variants of Delaunay
and alpha complexes, and code that does these computations is provided.
acknowledgement: "This project has received funding from the European Research\r\nCouncil
(ERC) under the European Union’s Horizon 2020 research and innovation\r\nprogramme,
grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund\r\n(FWF), grant
no. Z 342-N31, and from the DFG Collaborative Research Center TRR\r\n109, ‘Discretization
in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sebastiano
full_name: Cultrera di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera di Montesano
orcid: 0000-0001-6249-0832
- first_name: Ondrej
full_name: Draganov, Ondrej
id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
last_name: Draganov
orcid: 0000-0003-0464-3823
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic
alpha complexes. Foundations of Data Science. 2025;8:30-62. doi:10.3934/fods.2025003
apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian,
M. (2025). Chromatic alpha complexes. Foundations of Data Science. American
Institute of Mathematical Sciences. https://doi.org/10.3934/fods.2025003
chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner,
and Morteza Saghafian. “Chromatic Alpha Complexes.” Foundations of Data Science.
American Institute of Mathematical Sciences, 2025. https://doi.org/10.3934/fods.2025003.
ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian,
“Chromatic alpha complexes,” Foundations of Data Science, vol. 8. American
Institute of Mathematical Sciences, pp. 30–62, 2025.
ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2025. Chromatic
alpha complexes. Foundations of Data Science. 8, 30–62.
mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” Foundations
of Data Science, vol. 8, American Institute of Mathematical Sciences, 2025,
pp. 30–62, doi:10.3934/fods.2025003.
short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations
of Data Science 8 (2025) 30–62.
corr_author: '1'
date_created: 2025-11-02T23:01:33Z
date_published: 2025-03-01T00:00:00Z
date_updated: 2025-11-04T12:25:47Z
day: '01'
department:
- _id: HeEd
doi: 10.3934/fods.2025003
ec_funded: 1
external_id:
arxiv:
- '2212.03128'
intvolume: ' 8'
language:
- iso: eng
month: '03'
oa_version: Preprint
page: 30-62
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Foundations of Data Science
publication_identifier:
eissn:
- 2639-8001
publication_status: epub_ahead
publisher: American Institute of Mathematical Sciences
quality_controlled: '1'
related_material:
record:
- id: '15091'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Chromatic alpha complexes
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20657'
abstract:
- lang: eng
text: 'The Upper Bound Theorem for convex polytopes implies that the p-th Betti
number of the Čech complex of any set of N points in ℝ^d and any radius satisfies
β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions,
which prove that this upper bound is asymptotically tight. For example, we describe
a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number
of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of
the Čech complex at the other radius is n². '
acknowledgement: The first author is supported by the European Research Council (ERC),
grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian
Science Fund (FWF), grant no. I 02979-N35. The second author is supported by the
European Research Council (ERC), grant “GeoScape” and by the Hungarian Science Foundation
(NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian
Science Fund (FWF), grant no. Z 342-N31.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: János
full_name: Pach, János
id: E62E3130-B088-11EA-B919-BF823C25FEA4
last_name: Pach
citation:
ama: Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. Discrete
& Computational Geometry. 2025. doi:10.1007/s00454-025-00796-5
apa: Edelsbrunner, H., & Pach, J. (2025). Maximum Betti numbers of Čech complexes.
Discrete & Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-025-00796-5
chicago: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
Discrete & Computational Geometry. Springer Nature, 2025. https://doi.org/10.1007/s00454-025-00796-5.
ieee: H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” Discrete
& Computational Geometry. Springer Nature, 2025.
ista: Edelsbrunner H, Pach J. 2025. Maximum Betti numbers of Čech complexes. Discrete
& Computational Geometry.
mla: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
Discrete & Computational Geometry, Springer Nature, 2025, doi:10.1007/s00454-025-00796-5.
short: H. Edelsbrunner, J. Pach, Discrete & Computational Geometry (2025).
corr_author: '1'
date_created: 2025-11-19T09:44:58Z
date_published: 2025-11-10T00:00:00Z
date_updated: 2025-12-01T15:19:21Z
day: '10'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-025-00796-5
ec_funded: 1
external_id:
arxiv:
- '2310.14801'
isi:
- '001610592600001'
has_accepted_license: '1'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1007/s00454-025-00796-5
month: '11'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
publication: Discrete & Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
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relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Maximum Betti numbers of Čech complexes
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '20658'
abstract:
- lang: eng
text: The medial axis of a smoothly embedded surface in R^3 consists of all points
for which the Euclidean distance function on the surface has at least two global
minima. We generalize this notion to the mid-sphere axis, which consists of all
points for which the Euclidean distance function has two interchanging saddles
that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic
multi-scale approach to computing ridge-like structures on the surface. As a proof
of concept, an algorithm that computes stair-case approximations of the mid-sphere
axis is provided.
alternative_title:
- LNCS
article_processing_charge: No
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Martin H
full_name: Thoresen, Martin H
id: 47CB1472-F248-11E8-B48F-1D18A9856A87
last_name: Thoresen
citation:
ama: 'Edelsbrunner H, Stephenson ER, Thoresen MH. The mid-sphere cousin of the medial
axis transform. In: 4th International Joint Conference on Discrete Geometry
and Mathematical Morphology. Vol 16296. Springer Nature; 2025:133-147. doi:10.1007/978-3-032-09544-2_10'
apa: 'Edelsbrunner, H., Stephenson, E. R., & Thoresen, M. H. (2025). The mid-sphere
cousin of the medial axis transform. In 4th International Joint Conference
on Discrete Geometry and Mathematical Morphology (Vol. 16296, pp. 133–147).
Groningen, The Netherlands: Springer Nature. https://doi.org/10.1007/978-3-032-09544-2_10'
chicago: Edelsbrunner, Herbert, Elizabeth R Stephenson, and Martin H Thoresen. “The
Mid-Sphere Cousin of the Medial Axis Transform.” In 4th International Joint
Conference on Discrete Geometry and Mathematical Morphology, 16296:133–47.
Springer Nature, 2025. https://doi.org/10.1007/978-3-032-09544-2_10.
ieee: H. Edelsbrunner, E. R. Stephenson, and M. H. Thoresen, “The mid-sphere cousin
of the medial axis transform,” in 4th International Joint Conference on Discrete
Geometry and Mathematical Morphology, Groningen, The Netherlands, 2025, vol.
16296, pp. 133–147.
ista: 'Edelsbrunner H, Stephenson ER, Thoresen MH. 2025. The mid-sphere cousin of the medial
axis transform. 4th International Joint Conference on Discrete Geometry and Mathematical
Morphology. DGMM: Discrete Geometry and Mathematical Morphology, LNCS, vol. 16296,
133–147.'
mla: Edelsbrunner, Herbert, et al. “The Mid-Sphere Cousin of the Medial Axis Transform.”
4th International Joint Conference on Discrete Geometry and Mathematical Morphology,
vol. 16296, Springer Nature, 2025, pp. 133–47, doi:10.1007/978-3-032-09544-2_10.
short: H. Edelsbrunner, E.R. Stephenson, M.H. Thoresen, in:, 4th International Joint
Conference on Discrete Geometry and Mathematical Morphology, Springer Nature,
2025, pp. 133–147.
conference:
end_date: 2025-11-06
location: Groningen, The Netherlands
name: 'DGMM: Discrete Geometry and Mathematical Morphology'
start_date: 2025-11-03
date_created: 2025-11-23T23:01:37Z
date_published: 2025-11-01T00:00:00Z
date_updated: 2025-11-24T10:05:11Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/978-3-032-09544-2_10
external_id:
arxiv:
- '2504.14743'
intvolume: ' 16296'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2504.14743
month: '11'
oa: 1
oa_version: Preprint
page: 133-147
publication: 4th International Joint Conference on Discrete Geometry and Mathematical
Morphology
publication_identifier:
eissn:
- 1611-3349
isbn:
- '9783032095435'
issn:
- 0302-9743
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: The mid-sphere cousin of the medial axis transform
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 16296
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
_id: '20729'
abstract:
- lang: eng
text: 'Persistence modules (defined as a sequence of vector spaces and linear maps
between them) are a key tool in topological data analysis. They are easy to interpret
and fast to compute. However, when considering persistence maps (i.e. maps between
persistence modules), these properties are lost. We propose a new invariant for
persistence maps consisting of a partial matching such that: it is easy to interpret,
it is more discriminative than the image of the persistence map, and can be calculated
with cubical complexity.'
acknowledgement: Álvaro Torras-Casas contract is funded by the French Agence Nationale
de la Recherche through the project reference ANR-22-CPJ1-0047-01. Rocio Gonzalez-Diaz
is partially funded by the European Union under grant agreement no. 101070028-2
(REXASI-PRO).
article_processing_charge: Yes (in subscription journal)
author:
- first_name: Rocio
full_name: Gonzalez-Diaz, Rocio
last_name: Gonzalez-Diaz
- first_name: Manuel
full_name: Soriano Trigueros, Manuel
id: 15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8
last_name: Soriano Trigueros
orcid: 0000-0003-2449-1433
- first_name: Alvaro
full_name: Torras-Casas, Alvaro
last_name: Torras-Casas
citation:
ama: 'Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. Additive partial matchings
for persistent homology. In: Proceedings of the 2025 International Symposium
on Symbolic and Algebraic Computation. Association for Computing Machinery;
2025:188-196. doi:10.1145/3747199.3747561'
apa: 'Gonzalez-Diaz, R., Soriano Trigueros, M., & Torras-Casas, A. (2025). Additive
partial matchings for persistent homology. In Proceedings of the 2025 International
Symposium on Symbolic and Algebraic Computation (pp. 188–196). Guanajuato,
Mexico: Association for Computing Machinery. https://doi.org/10.1145/3747199.3747561'
chicago: Gonzalez-Diaz, Rocio, Manuel Soriano Trigueros, and Alvaro Torras-Casas.
“Additive Partial Matchings for Persistent Homology.” In Proceedings of the
2025 International Symposium on Symbolic and Algebraic Computation, 188–96.
Association for Computing Machinery, 2025. https://doi.org/10.1145/3747199.3747561.
ieee: R. Gonzalez-Diaz, M. Soriano Trigueros, and A. Torras-Casas, “Additive partial
matchings for persistent homology,” in Proceedings of the 2025 International
Symposium on Symbolic and Algebraic Computation, Guanajuato, Mexico, 2025,
pp. 188–196.
ista: 'Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. 2025. Additive partial
matchings for persistent homology. Proceedings of the 2025 International Symposium
on Symbolic and Algebraic Computation. ISSAC: International Symposium on Symbolic
and Algebraic Computation, 188–196.'
mla: Gonzalez-Diaz, Rocio, et al. “Additive Partial Matchings for Persistent Homology.”
Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation,
Association for Computing Machinery, 2025, pp. 188–96, doi:10.1145/3747199.3747561.
short: R. Gonzalez-Diaz, M. Soriano Trigueros, A. Torras-Casas, in:, Proceedings
of the 2025 International Symposium on Symbolic and Algebraic Computation, Association
for Computing Machinery, 2025, pp. 188–196.
conference:
end_date: 2025-08-01
location: Guanajuato, Mexico
name: 'ISSAC: International Symposium on Symbolic and Algebraic Computation'
start_date: 2025-07-28
corr_author: '1'
date_created: 2025-12-07T23:02:01Z
date_published: 2025-11-10T00:00:00Z
date_updated: 2025-12-09T13:46:42Z
day: '10'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1145/3747199.3747561
file:
- access_level: open_access
checksum: 1c299cca165a20e2518afe4fda63dbf1
content_type: application/pdf
creator: dernst
date_created: 2025-12-09T13:43:17Z
date_updated: 2025-12-09T13:43:17Z
file_id: '20751'
file_name: 2025_ISSAC_GonzalezDiaz.pdf
file_size: 761617
relation: main_file
success: 1
file_date_updated: 2025-12-09T13:43:17Z
has_accepted_license: '1'
language:
- iso: eng
month: '11'
oa: 1
oa_version: Published Version
page: 188-196
publication: Proceedings of the 2025 International Symposium on Symbolic and Algebraic
Computation
publication_identifier:
isbn:
- '9798400720758'
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
scopus_import: '1'
status: public
title: Additive partial matchings for persistent homology
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
_id: '13182'
abstract:
- lang: eng
text: "We characterize critical points of 1-dimensional maps paired in persistent
homology\r\ngeometrically and this way get elementary proofs of theorems about
the symmetry\r\nof persistence diagrams and the variation of such maps. In particular,
we identify\r\nbranching points and endpoints of networks as the sole source of
asymmetry and\r\nrelate the cycle basis in persistent homology with a version
of the stable marriage\r\nproblem. Our analysis provides the foundations of fast
algorithms for maintaining a\r\ncollection of sorted lists together with its persistence
diagram."
acknowledgement: Open access funding provided by Austrian Science Fund (FWF). This
project has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme, grant no. 788183,
from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and
from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry
and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35. The authors of
this paper thank anonymous reviewers for their constructive criticism and Monika
Henzinger for detailed comments on an earlier version of this paper.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Ranita
full_name: Biswas, Ranita
id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
last_name: Biswas
orcid: 0000-0002-5372-7890
- first_name: Sebastiano
full_name: Cultrera Di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera Di Montesano
orcid: 0000-0001-6249-0832
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization
of the persistence of 1D maps. Journal of Applied and Computational Topology.
2024;8:1101-1119. doi:10.1007/s41468-023-00126-9
apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M.
(2024). Geometric characterization of the persistence of 1D maps. Journal of
Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-023-00126-9
chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.”
Journal of Applied and Computational Topology. Springer Nature, 2024. https://doi.org/10.1007/s41468-023-00126-9.
ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric
characterization of the persistence of 1D maps,” Journal of Applied and Computational
Topology, vol. 8. Springer Nature, pp. 1101–1119, 2024.
ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Geometric
characterization of the persistence of 1D maps. Journal of Applied and Computational
Topology. 8, 1101–1119.
mla: Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D
Maps.” Journal of Applied and Computational Topology, vol. 8, Springer
Nature, 2024, pp. 1101–19, doi:10.1007/s41468-023-00126-9.
short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal
of Applied and Computational Topology 8 (2024) 1101–1119.
corr_author: '1'
date_created: 2023-07-02T22:00:44Z
date_published: 2024-10-01T00:00:00Z
date_updated: 2026-04-07T12:58:47Z
day: '01'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/s41468-023-00126-9
ec_funded: 1
external_id:
pmid:
- '39678706'
file:
- access_level: open_access
checksum: d493df5088c222b88d9ca46b623ad0ee
content_type: application/pdf
creator: dernst
date_created: 2025-01-09T07:39:41Z
date_updated: 2025-01-09T07:39:41Z
file_id: '18783'
file_name: 2024_JourApplCompTopo_Biswas.pdf
file_size: 476896
relation: main_file
success: 1
file_date_updated: 2025-01-09T07:39:41Z
has_accepted_license: '1'
intvolume: ' 8'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
page: 1101-1119
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316
grant_number: I4887
name: Persistent Homology, Algorithms and Stochastic Geometry
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '15094'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Geometric characterization of the persistence of 1D maps
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2024'
...
---
_id: '14345'
abstract:
- lang: eng
text: For a locally finite set in R2, the order-k Brillouin tessellations form an
infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely
dense and generic, then the corresponding infinite sequences of minimum and maximum
angles are both monotonic in k. As an example, a stationary Poisson point process
in R2 is locally finite, coarsely dense, and generic with probability one. For
such a set, the distributions of angles in the Voronoi tessellations, Delaunay
mosaics, and Brillouin tessellations are independent of the order and can be derived
from the formula for angles in order-1 Delaunay mosaics given by Miles (Math.
Biosci. 6, 85–127 (1970)).
acknowledgement: Work by all authors but A. Garber is supported by the European Research
Council (ERC), Grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund
(FWF), Grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,
Austrian Science Fund (FWF), Grant No. I 02979-N35. Work by A. Garber is partially
supported by the Alexander von Humboldt Foundation.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Alexey
full_name: Garber, Alexey
last_name: Garber
- first_name: Mohadese
full_name: Ghafari, Mohadese
last_name: Ghafari
- first_name: Teresa
full_name: Heiss, Teresa
id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
last_name: Heiss
orcid: 0000-0002-1780-2689
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher
order Brillouin tessellations and related tilings in the plane. Discrete and
Computational Geometry. 2024;72:29-48. doi:10.1007/s00454-023-00566-1
apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2024).
On angles in higher order Brillouin tessellations and related tilings in the plane.
Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-023-00566-1
chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and
Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related
Tilings in the Plane.” Discrete and Computational Geometry. Springer Nature,
2024. https://doi.org/10.1007/s00454-023-00566-1.
ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles
in higher order Brillouin tessellations and related tilings in the plane,” Discrete
and Computational Geometry, vol. 72. Springer Nature, pp. 29–48, 2024.
ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles
in higher order Brillouin tessellations and related tilings in the plane. Discrete
and Computational Geometry. 72, 29–48.
mla: Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations
and Related Tilings in the Plane.” Discrete and Computational Geometry,
vol. 72, Springer Nature, 2024, pp. 29–48, doi:10.1007/s00454-023-00566-1.
short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete
and Computational Geometry 72 (2024) 29–48.
corr_author: '1'
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