On the Beer index of convexity and its variants

Abstract

Let S be a subset of R d Rd with finite positive Lebesgue measure. The Beer index of convexityb(S) b(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) 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. We show that every set S⊆R 2 S⊆R2 with simply connected components satisfies b(S)⩽αc(S) b(S)⩽αc(S) for an absolute constant α α , provided b(S) 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) b(S) . For 1⩽k⩽d 1⩽k⩽d , the k-index of convexityb k (S) bk(S) of a set S⊆R d S⊆Rd is the probability that the convex hull of a (k+1) (k+1) -tuple of points chosen uniformly independently at random from S is contained in S. We show that for every d⩾2 d⩾2 there is a constant β(d)>0 β(d)>0 such that every set S⊆R d S⊆Rd satisfies b d (S)⩽βc(S) bd(S)⩽βc(S) , provided b d (S) bd(S) exists. We provide an almost matching lower bound by showing that there is a constant γ(d)>0 γ(d)>0 such that for every ε∈(0,1) ε∈(0,1) there is a set S⊆R d S⊆Rd of Lebesgue measure 1 satisfying c(S)⩽ε c(S)⩽ε and b d (S)⩾γεlog 2 1/ε ⩾γc(S)log 2 1/c(S) bd(S)⩾γεlog2⁡1/ε⩾γc(S)log2⁡1/c(S)