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Approximate Carathéodory’s Theorem in Uniformly Smooth Banach Spaces

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Abstract

We study the ‘no-dimension’ analogue of Carathéodory’s theorem in Banach spaces. We prove such a result together with its colorful version for uniformly smooth Banach spaces. It follows that uniform smoothness leads to a greedy de-randomization of Maurey’s classical lemma Pisier (in: Séminaire Analyse fonctionnelle (dit “Maurey-Schwartz”), 1980), which is itself a ‘no-dimension’ analogue of Carathéodory’s theorem with a probabilistic proof. We find the asymptotically tight upper bound on the deviation of the convex hull from the k-convex hull of a bounded set in \(L_p\) with \(1 < p \le 2\) and get asymptotically the same bound as in Maurey’s lemma for \(L_p\) with \(1< p < \infty \).

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Acknowledgements

I wish to thank Imre Bárány for bringing the problem to my attention.

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Authors and Affiliations

  1. Department of Mathematics, University of Fribourg, Chemin du Musée 23, 1700, Fribourg, Switzerland

    Grigory Ivanov

  2. Department of Higher Mathematics, Moscow Institute of Physics and Technology, Institutskii pereulok 9, Dolgoprudny, Moscow Region, Russia, 141700

    Grigory Ivanov

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  1. Grigory Ivanov
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Correspondence to Grigory Ivanov.

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Supported by the Swiss National Science Foundation Grant 200021_179133. Supported by the Russian Foundation for Basic Research, Project 18-01-00036 A.

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Ivanov, G. Approximate Carathéodory’s Theorem in Uniformly Smooth Banach Spaces. Discrete Comput Geom 66, 273–280 (2021). https://doi.org/10.1007/s00454-019-00130-w

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