Abstract
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 a relationship between these two natural measures of convexity of S.
We show that every subset S of the plane with simply connected components satisfies b(S) LESS-THAN OR EQUAL TO alpha c(S) for an absolute constant alpha, 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 LESS-THAN OR EQUAL TO k LESS-THAN OR EQUAL TO d, the k-index of convexity b_k(S) of a subset S of 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 GREATER-THAN OR EQUAL TO 2 there is a constant beta(d) > 0 such that every subset S of R^d satisfies b_d(S) {= beta c(S), provided 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 epsilon from (0,1] there is a subset S of R^d of Lebesgue measure one satisfying c(S) LESS-THAN OR EQUAL TO epsilon and b_d(S) GREATER-THAN OR EQUAL TO (gamma epsilon)/log_2(1/epsilon) GREATER-THAN OR EQUAL TO (gamma c(S))/log_2(1/c(S)).