  Let $\Omega$ be a bounded closed convex set in ${\mathbb R}^d$ with non-empty
interior, and let ${\cal C}_r(\Omega)$ be the class of convex functions on
$\Omega$ with $L^r$-norm bounded by $1$. We obtain sharp estimates of the
$\epsilon$-entropy of ${\cal C}_r(\Omega)$ under $L^p(\Omega)$ metrics, $1\le
p<r\le \infty$. In particular, the results imply that the universal lower bound
$\epsilon^{-d/2}$ is also an upper bound for all $d$-polytopes, and the
universal upper bound of $\epsilon^{-\frac{(d-1)}{2}\cdot \frac{pr}{r-p}}$ for
$p>\frac{dr}{d+(d-1)r}$ is attained by the closed unit ball. While a general
convex body can be approximated by inscribed polytopes, the entropy rate does
not carry over to the limiting body. Our results have applications to questions
concerning rates of convergence of nonparametric estimators of high-dimensional
shape-constrained functions.
