Metric Entropy of Some Classes of Sets with Differentiable Boundaries
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 .
KeywordsStochastic Process Probability Theory Lower Order Statistical Theory Lebesgue Measure
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