Depth in arrangements: Dehn–Sommerville–Euler relations with applications

Biswas, Ranita; Cultrera Di Montesano, Sebastiano; Edelsbrunner, Herbert; Saghafian, Morteza

Depth in arrangements: Dehn–Sommerville–Euler relations with applications

Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. 8, 557–578.

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DOI

10.1007/s41468-024-00173-w

Journal Article | Published | English

Scopus indexed

Author

Biswas, Ranita^ISTA ; Cultrera di Montesano, Sebastiano^ISTA ; Edelsbrunner, Herbert^ISTA ; Saghafian, Morteza^ISTA

Corresponding author has ISTA affiliation

Department

Edelsbrunner Group

Grant

Alpha Shape Theory Extended
Mathematics, Computer Science
Persistence and stability of geometric complexes

Abstract

The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.

Publishing Year

2024

Date Published

2024-09-01

Journal Title

Journal of Applied and Computational Topology

Publisher

Springer Nature

Acknowledgement

The authors thank Uli Wagner and Emo Welzl for comments on an earlier version of this paper, and for pointing out related work in the prior literature. Open access funding provided by Institute of Science and Technology (IST Austria). 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.

Volume

Page

557-578

ISSN

2367-1726

eISSN

2367-1734

IST-REx-ID

15380

Cite this

Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. 2024;8:557-578. doi:10.1007/s41468-024-00173-w

Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2024). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-024-00173-w

Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Journal of Applied and Computational Topology. Springer Nature, 2024. https://doi.org/10.1007/s41468-024-00173-w.

R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” Journal of Applied and Computational Topology, vol. 8. Springer Nature, pp. 557–578, 2024.

Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Journal of Applied and Computational Topology, vol. 8, Springer Nature, 2024, pp. 557–78, doi:10.1007/s41468-024-00173-w.

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