On the Beer Index of Convexity and Its Variants

Abstract

Let S be a subset of mathbb {R}^d with finite positive Lebesgue measure. The Beer index of convexity{text {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{text {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 Ssubseteq mathbb {R}^2 with simply connected components satisfies {text {b}}(S)leqslant alpha {text {c}}(S) for an absolute constant alpha , provided {text {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 {text {b}}(S). For 1leqslant kleqslant d, the k-index of convexity{text {b}}_k(S) of a set Ssubseteq mathbb {R}^d 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 dgeqslant 2 there is a constant beta (d)>0 such that every set Ssubseteq mathbb {R}^d satisfies {text {b}}_d(S)leqslant beta {text {c}}(S), provided {text {b}}_d(S) exists. We provide an almost matching lower bound by showing that there is a constant gamma (d)>0 such that for every varepsilon in (0,1) there is a set Ssubseteq mathbb {R}^d of Lebesgue measure 1 satisfying {text {c}}(S)leqslant varepsilon and {text {b}}_d(S)geqslant gamma frac{varepsilon }{log _2{1/varepsilon }}geqslant gamma frac{{text {c}}(S)}{log _2{1/{text {c}}(S)}}.