Metric Entropy of Some Classes of Sets with Differentiable Boundaries

Abstract

Let I(k, α, M) be the class of all subsets A of R ^k whose boundaries are given by functions from the sphere S ^{k −1} into R ^k with derivatives of order % α, all bounded by M. (The precise definition, for all α > 0, involves Hölder conditions.) Let N _d (ε) be the minimum number of sets required to approximate every set in I(k, α, M) within ε for the metric d, which is the Hausdorff metric h or the Lebesgue measure of the symmetric difference, d _λ . It is shown that up to factors of lower order of growth, N _d (ε) can be approximated by exp(ε ^−r) as ε ↓ 0, where r = (k − 1)/α if d = h or if d = d _λ and α > 1. For d = d _λ and (k − 1)/k < α < 1, r < (k − 1)/(kα − k + 1). The proof uses results of A. N. Kolmogorov and V. N. Tikhomirov [4].

This research was partially supported by National Science Foundation Grant GP-29072.