Let Ω be a bounded closed convex set in ℝ with non-empty interior, and let 𝒞 (Ω) be the class of convex functions on Ω with -norm bounded by 1. We obtain sharp estimates of the -entropy of 𝒞 (Ω) under (Ω) metrics, 1 ≤ < ≤ ∞. In particular, the results imply that the universal lower bound is also an upper bound for all -polytopes, and the universal upper bound of [Formula: see text] for [Formula: see text] 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.

To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of . For restrictions to the Euclidean ball in odd dimensions, to the rotation group , and to the Grassmannian manifold , we compute the kernels' Fourier coefficients and determine their asymptotics. The -discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For , the nonequispaced fast Fourier transform is publicly available, and, for , the transform is derived here. We also provide numerical experiments for and .