On the Beer Index of Convexity and Its Variants

Let S be a subset of R^d 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⊆ 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 d⩾ 2 there is a constant β(d) > 0 such that every set S⊆ R^d 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⊆ R^d of Lebesgue measure 1 satisfying c (S) ⩽ ε and bd(S)⩾γεlog21/ε⩾γc(S)log21/c(S).