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

TL;DR

This paper extends Carathéodory's theorem to uniformly smooth Banach spaces, providing new bounds and derandomization techniques that improve understanding of convex hull approximations in these spaces.

Contribution

It proves a no-dimension analogue of Carathéodory's theorem for uniformly smooth Banach spaces and derives tight bounds for convex hull deviations in Lp spaces.

Findings

Established a no-dimension Carathéodory-type theorem for uniformly smooth Banach spaces.

Derived asymptotically tight bounds on convex hull deviations in Lp spaces.

Provided a greedy derandomization of Maurey's lemma based on uniform smoothness.

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 $1 < p < \infty.$