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arXiv · 1904.06729

Approximate Carath{\'e}odory's theorem in uniformly smooth Banach spaces

G. M. Ivanov
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Abstract

We study the 'no-dimension' analogue of Carath{\'e}odory'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 \cite{pisier1980remarques}, which is itself a 'no-dimension' analogue of Carath{\'e}odory'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 $1 < p \leq 2$ and get asymptotically the same bound as in Maurey's lemma for $L_p$ with $1 < p < \infty.$

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G. M. Ivanov. 2019-04-14. Approximate Carath{\'e}odory's theorem in uniformly smooth Banach spaces. https://arxiv.org/abs/1904.06729

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