On the Beer Index of Convexity and Its Variants

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 the relationship between these two natural measures of convexity.We show that every set S ⊆ R 2 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. that this estimate holds for simple polygons.We also consider higher-order generalizations of b(S). For 1 k d, the k-index of convexity b 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) β c(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 b d (S) γ ε log 2 1/ε

Peeling Potatoes Near-Optimally in Near-Linear Time

“…Also, it is a key contribution of this paper to realize that such connection between the probability of being co-visible and the area of the largest convex body could exist. This result has been improved by Balko et al [5]. Using their new result slightly improves the final running time of our algorithms.…”

“…For each constant ε, the bottleneck in the running time of our algorithm is here, in our use of Theorem 13 to compute the visibility graph. With the improvement of Balko et al [5], stated in Theorem 11, all other steps can be made to run in time O(n log n log(1/δ)) (for constant ε).…”

“…Consider one of the iterations of the repeat loop (lines [5][6][7][8][9][10][11][12][13][14]. A slight modification of the proof of Lemma 12 gives that the expected size of the visibility graph G(P, R) is bounded by E[|E(G(P, R))|] = r 2 · Pr[two random points are visible in P ]…”

“…Such relation was unknown and we believe that it is of independent interest. See Theorems 9, 10 and the follow up work [5] (summarized in Theorem 11 here) for the precise relations. Perimeter.…”

The convexification effect of Minkowski summation

“…For dimension 2, this has been studied, for example, by Goodman [40]. Balko et al [9] discuss this notion in general dimension, and also give some inequalities relating the convexity ratio and Beer's index of convexity. Once again, to get a measure of non-convexity, it is more natural to consider κ(A) = 1 − Vol n (L(A)) Vol n (A) ,…”