Entropy of convex functions on $R^d$

Abstract

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.