Metric Entropy of Convex Hulls in Hilbert Spaces |
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Authors: | Carl Bernd |
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Institution: | Universität Jena, Fakultät für Mathematik und Informatik D-07740 Jena, Germany |
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Abstract: | We show in this note the following statement which is an improvementover a result of R. M. Dudley and which is also of independentinterest. Let X be a set of a Hilbert space with the propertythat there are constants , >0, and for each n N, the setX can be covered by at most n balls of radius n. Then,for each n N, the convex hull of X can be covered by 2n ballsof radius . The estimate is best possible for all n N, apart from the value c=c(, , X).In other words, let N(, X), >0, be the minimal number ofballs of radius covering the set X. Then the above result isequivalent to saying that if N(, X)=O(1/) as 0, thenfor the convex hull conv (X) of X, N(, conv (X)) =O(exp(2/(12))). Moreover, we give an interplay between several coveringparameters based on coverings by balls (entropy numbers) andcoverings by cylindrical sets (Kolmogorov numbers). 1991 MathematicsSubject Classification 41A46. |
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