On the Beer Index of Convexity and Its Variants

Martin Balko, Vít Jelínek, Pavel Valtr, Bartosz Walczak

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


Let S be a subset of Rd with finite positive Lebesgue measure. The Beer index of convexityb (S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratioc (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. We show that every set S⊆ R2 with simply connected components satisfies b (S) ⩽ αc (S) for an absolute constant α, provided b (S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. that this estimate holds for simple polygons. We also consider higher-order generalizations of b (S). For 1 ⩽ k⩽ d, the k-index of convexityb k(S) of a set S⊆ Rd is the probability that the convex hull of a (k+ 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 b d(S) ⩽ βc (S) , provided b d(S) exists. We provide an almost matching lower bound by showing that there is a constant γ(d) > 0 such that for every ε∈ (0 , 1) there is a set S⊆ Rd of Lebesgue measure 1 satisfying c (S) ⩽ ε and bd(S)⩾γεlog21/ε⩾γc(S)log21/c(S).

Original languageEnglish
Pages (from-to)179-214
Number of pages36
JournalDiscrete and Computational Geometry
Issue number1
StatePublished - 1 Jan 2017
Externally publishedYes


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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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