Metric entropy of some classes of sets with differentiable boundaries

AbstractLet I(k, α, M) be the class of all subsets A of Rk whose boundaries are given by functions from the sphere Sk − 1 into Rk with derivatives of order ⩽α, all bounded by M. (The precise definition, for all α > 0, involves Hölder conditions.) Let Nd(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, dλ. It is shown that up to factors of lower order of growth, Nd(ϵ) can be approximated by exp(ϵ−ras ϵ ↓ 0, where r = (k − 1)αif d = h or if d = dλand α ⩾ 1. For d = dλand(k − 1)k < α ⩽ 1, r ⩽ (k − 1)(kα − k + 1). The proof uses results of A. N. Kolmogorov and V. N. Tikhomirov [4]