  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
$\operatorname{b}_d(S)\geq\gamma\frac{\varepsilon}{\log_2{1/\varepsilon}}\geq\gamma\frac{\operatorname{c}(S)}{\log_2{1/\operatorname{c}(S)}}$.
