Selected Works of R.M. Dudley pp 433-443 | Cite as

# 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].

## Keywords

Stochastic Process Probability Theory Lower Order Statistical Theory Lebesgue Measure## Preview

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## References

- 1.T. Bonnesen andW. Fenchel, “Theorie der Konvexen Körper,” Springer Verlag, Berlin, 1934.Google Scholar
- 2.G. F. Clements, Entropies of several sets of real valued functions,
*Pacific J. Math*.**13**(1963), 1085–1095.MATHMathSciNetGoogle Scholar - 3.R. M. Dudley, “Sample functions of the Gaussian process,”
*Annals of Probability***1**(1973), 66–103.MATHCrossRefMathSciNetGoogle Scholar - 4.A. N. Kolmogorov and V. M. Tikhomirov,
*ε*-entropy and*ε*-capacity of sets in functional spaces,*Amer. Math. Soc. Transl*. (*Ser. 2*)**17**(1961), 277–364 (from*Uspekhi Mat. Nauk***14**(1959), 3–86).Google Scholar - 5.M. Loève, “Probability Theory,” Van Nostrand, Princeton, N.J., 1963.Google Scholar
- 6.G. G. Lorentz, Metric entropy and approximation,
*Bull. Amer. Math. Soc*.**72**(1966), 903–937.MATHCrossRefMathSciNetGoogle Scholar