Let $S$ be a subset of $\mathbb{ R}^d$ with finite positive Lebesgue measure. The Beer index of convexity $\operatorname{ b}(S)$ of $S$ is the probability that two points of $S$ chosen uniformly independently at random see each other in $S$. The convexity ratio $\operatorname{ c}(S)$ 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 $\operatorname{ b}(S)\leq\alpha\operatorname{ c}(S)$ for an absolute constant $\alpha$, provided $\operatorname{ b}(S)$ 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 $\operatorname{ b}(S)$. For $1\leq k\leq d$, the $k$-index of convexity $\operatorname{ b}_k(S)$ 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 $d\geq 2$ there is a constant $\beta(d)>0$ such that every set $S\subseteq\mathbb{ R}^d$ satisfies $\operatorname{ b}_d(S)\leq\beta\operatorname{ c}(S)$, provided $\operatorname{ b}_d(S)$ 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 $\operatorname{ c}(S)\leq\varepsilon$ and $\operatorname{ b}_d(S)\geq\gamma\frac{ \varepsilon}{ \log_2{ 1/\varepsilon}}\geq\gamma\frac{ \operatorname{ c}(S)}{ \log_2{ 1/\operatorname{ c}(S)}}$.
Categories: math.MG cs.CG math.CO