---
res:
bibo_abstract:
- 'We introduce a new variant of quantitative Helly-type theorems: the minimal homothetic
distance of the intersection of a family of convex sets to the intersection of
a subfamily of a fixed size. As an application, we establish the following quantitative
Helly-type result for the diameter. If $K$ is the intersection of finitely many
convex bodies in $\mathbb{R}^d$, then one can select $2d$ of these bodies whose
intersection is of diameter at most $(2d)^3{diam}(K)$. The best previously known
estimate, due to Brazitikos [Bull. Hellenic Math. Soc., 62 (2018), pp. 19--25],
is $c d^{11/2}$. Moreover, we confirm that the multiplicative factor $c d^{1/2}$
conjectured by Bárány, Katchalski, and Pach [Proc. Amer. Math. Soc., 86 (1982),
pp. 109--114] cannot be improved. The bounds above follow from our key result
that concerns sparse approximation of a convex polytope by the convex hull of
a well-chosen subset of its vertices: Assume that $Q \subset {\mathbb R}^d$ is
a polytope whose centroid is the origin. Then there exist at most 2d vertices
of $Q$ whose convex hull $Q^{\prime \prime}$ satisfies $Q \subset - 8d^3 Q^{\prime
\prime}.$@eng'
bibo_authorlist:
- foaf_Person:
foaf_givenName: Grigory
foaf_name: Ivanov, Grigory
foaf_surname: Ivanov
foaf_workInfoHomepage: http://www.librecat.org/personId=87744F66-5C6F-11EA-AFE0-D16B3DDC885E
- foaf_Person:
foaf_givenName: Marton
foaf_name: Naszodi, Marton
foaf_surname: Naszodi
bibo_doi: 10.1137/21M1403308
bibo_issue: '2'
bibo_volume: 36
dct_date: 2022^xs_gYear
dct_identifier:
- UT:000793158200002
dct_isPartOf:
- http://id.crossref.org/issn/0895-4801
dct_language: eng
dct_publisher: Society for Industrial and Applied Mathematics@
dct_title: 'A quantitative Helly-type theorem: Containment in a homothet@'
...