Metric Entropy of Some Classes of Sets with Differentiable Boundaries

  • R. M. Dudley
Part of the Selected Works in Probability and Statistics book series (SWPS)


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)/(k + 1). The proof uses results of A. N. Kolmogorov and V. N. Tikhomirov [4].


Stochastic Process Probability Theory Lower Order Statistical Theory Lebesgue Measure 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.Mathematics DepartmentMassachusetts Institute of TechnologyCambridgeUSA

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