Metric entropy of convex hulls in Hilbert spaces

Wenbo V. Li; Werner Linde

ArticleOriginal scientific text

Title

Metric entropy of convex hulls in Hilbert spaces

Authors ¹, ²

Affiliations

Department of Mathematics, University of Delaware Newark, DE 19711, U.S.A.
Institut für Stochastik, FSU Jena Ernst-Abbe-Platz 1-4, 07743 Jena, Germany.

Abstract

Let T be a precompact subset of a Hilbert space. We estimate the metric entropy of co(T), the convex hull of T, by quantities originating in the theory of majorizing measures. In a similar way, estimates of the Gelfand width are provided. As an application we get upper bounds for the entropy of co(T),

T = {t_{1}, t_{2}, ...}

,

| | t_{j} | | \leq a_{j}

, by functions of the

a_{j}

's only. This partially answers a question raised by K. Ball and A. Pajor (cf. [1]). Our estimates turn out to be optimal in the case of slowly decreasing sequences

{(a_{j})}_{j = 1}^{\infty}

.

Keywords

ENG

metric entropy, convex hull, majorizing measure, Gaussian process

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Title

Metric entropy of convex hulls in Hilbert spaces

Affiliations

Abstract

Keywords

Bibliography