Randomized series and Geometry of Banach spaces

TL;DR

This paper explores the properties of randomized series and their implications for the geometric structure of Banach spaces, establishing new criteria for convexity and monotonicity in these spaces.

Contribution

It introduces novel criteria linking convexity properties of Banach spaces with their p-convexifications, advancing understanding of their geometric structure.

Findings

$ ext{ell}_ ext{infty}^n$ is representable in $X$ iff in $L_p(X)$

A Banach lattice $E$ is uniformly monotone iff $E^{(p)}$ is uniformly convex

A K"othe function space $E$ is upper locally uniformly monotone iff $E^{(p)}$ is midpoint locally uniformly convex

Abstract

We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice $E$ is uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is uniformly convex and that a K\"othe function space $E$ is upper locally uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is midpoint locally uniformly convex.