摘要：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) = 2 there is a constant beta(d) > 0 such that every subset S of R^d satisfies b_d(S) 0 such that for every epsilon from (0,1] there is a subset S of R^d of Lebesgue measure one satisfying c(S) = (gamma epsilon)/log_2(1/epsilon) >= (gamma c(S))/log_2(1/c(S)).
关键词：Beer index of convexity; convexity ratio; convexity measure; visibility
