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Metric entropy of convex hulls in Hilbert spaces

Wenbo Li; Werner Linde — 2000

Studia Mathematica

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

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Metric entropy of convex hulls in Hilbert spaces

A Gaussian correlation inequality and its applications to small ball probabilities.

Expected lengths of minimum spanning trees for non-identical edge distributions.

Large deviations for local times of stable processes and stable random walks in 1 dimension.

Hypercontractivity and comparison of moments of iterated maxima and minima of independent random variables.