On the Beer Index of Convexity and Its Variants

Discrete & Computational Geometry, Sep 2016

Let S be a subset of $\mathbb {R}^d$ with finite positive Lebesgue measure. The Beer index of convexity ${\text {b}}(S)$ of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio ${\text {c}}(S)$ 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 $S\subseteq \mathbb {R}^2$ with simply connected components satisfies ${\text {b}}(S)\leqslant \alpha {\text {c}}(S)$ for an absolute constant $\alpha$, provided ${\text {b}}(S)$ 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 ${\text {b}}(S)$. For $1\leqslant k\leqslant d$, the k-index of convexity ${\text {b}}_k(S)$ of a set $S\subseteq \mathbb {R}^d$ 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 $d\geqslant 2$ there is a constant $\beta (d)>0$ such that every set $S\subseteq \mathbb {R}^d$ satisfies ${\text {b}}_d(S)\leqslant \beta {\text {c}}(S)$, provided ${\text {b}}_d(S)$ exists. We provide an almost matching lower bound by showing that there is a constant $\gamma (d)>0$ such that for every $\varepsilon \in (0,1)$ there is a set $S\subseteq \mathbb {R}^d$ of Lebesgue measure 1 satisfying ${\text {c}}(S)\leqslant \varepsilon$ and ${\text {b}}_d(S)\geqslant \gamma \frac{\varepsilon }{\log _2{1/\varepsilon }}\geqslant \gamma \frac{{\text {c}}(S)}{\log _2{1/{\text {c}}(S)}}$.

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Martin Balko, Vít Jelínek, Pavel Valtr, Bartosz Walczak. On the Beer Index of Convexity and Its Variants, Discrete & Computational Geometry, 2017, 179-214, DOI: 10.1007/s00454-016-9821-3