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Metric entropy of convex hulls in Hilbert spaces

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Li W. V., Linde W.

Studia Mathematica

2000

tom 139

nr 1

29-45

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), $T={t_1,t_2,...}$, $||t_j||≤a_j$, 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 $(a_j)_{j=1}^∞$.

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1 Studia Mathematica

1 Li W. V.

1 Linde W.

1 2000

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Metric entropy of convex hulls in Hilbert spaces