Entropy of Convex Functions on

摘要

Let be a bounded closed convex set in with nonempty interior, and let be the class of convex functions on with -norm bounded by 1. We obtain sharp estimates of the -entropy of under metrics, . In particular, the results imply that the universal lower bound is also an upper bound for all d-polytopes, and the universal upper bound of for 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.