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Kuelbs–Li Inequalities and Metric Entropy of Convex Hulls

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Let T be a precompact subset of a Hilbert space. We make use of a precise link between the absolutely convex hull \(\operatorname{aco}(T)\) and the reproducing kernel Hilbert space of a Gaussian random variable constructed from T. Firstly, we avail ourselves of it for optimality considerations concerning the well-known Kuelbs–Li inequalities. Secondly, this enables us to apply small deviation results to the problem of estimating the metric entropy of \(\operatorname{aco}(T)\) in dependence of the metric entropy of T.

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Acknowledgement

The author wants to express his gratitude to Prof. W. Linde for his support over the last years including all the interesting discussions about the topic. He thanks Prof. M.A. Lifshits for the permission to include the proof of Proposition 8 and for an impulse to Proposition 5 and Dr. F. Aurzada for careful reading of the manuscript while giving many useful comments and suggestions.

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  1. Institut für Stochastik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743, Jena, Germany

    Oliver Kley

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  1. Oliver Kley
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Kley, O. Kuelbs–Li Inequalities and Metric Entropy of Convex Hulls. J Theor Probab 26, 649–665 (2013). https://doi.org/10.1007/s10959-012-0408-5

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