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 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 ⊆ ℝ^d 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 ⊆ 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 ε (0, 1] there is a set S ⊆ ℝ^d of Lebesgue measure one satisfying c(S) ≤ ⊆ and (eqution presented).