On the Beer Index of Convexity and Its Variants

Martin Balko, Vít Jelínek, Pavel Valtr, Bartosz Walczak

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Let S be a subset of ℝd with finite positive Lebesgue measure. The Beer index of convexity b(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio c(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate the relationship between these two natural measures of convexity of S. We show that every set S ⊆ ℝ2 with simply connected components satisfies b(S) ≤ α c(S) for an absolute constant α, provided b(S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. asserting that this estimate holds for simple polygons. We also consider higher-order generalizations of b(S). For 1 ≤ κ ≤ d, the k-index of convexity bk(S) of S ⊆ ℝd is the probability that the convex hull of a (κ+1)-tuple of points chosen uniformly independently at random from S is contained in S. We show that for every d>2 there is a constant β(d)>0 such that every set S ⊆ ℝd satisfies bd(S) ≤ βc(S), provided bd(S) exists. We provide an almost matching lower bound by showing that there is a constant γ(d)>0 such that for every ε ε (0, 1] there is a set S ⊆ ℝd of Lebesgue measure one satisfying c(S) ≤ ⊆ and (eqution presented).

Original languageEnglish
Title of host publication31st International Symposium on Computational Geometry, SoCG 2015
EditorsJanos Pach, Janos Pach, Lars Arge
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages406-420
Number of pages15
ISBN (Electronic)9783939897835
DOIs
StatePublished - 1 Jun 2015
Externally publishedYes
Event31st International Symposium on Computational Geometry, SoCG 2015 - Eindhoven, Netherlands
Duration: 22 Jun 201525 Jun 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume34
ISSN (Print)1868-8969

Conference

Conference31st International Symposium on Computational Geometry, SoCG 2015
Country/TerritoryNetherlands
CityEindhoven
Period22/06/1525/06/15

Keywords

  • Beer index of convexity
  • Convexity measure
  • Convexity ratio
  • Visibility

ASJC Scopus subject areas

  • Software

Cite this