Roundness Properties of Banach spaces

TL;DR

This paper investigates the roundness and coroundness properties of Banach spaces, providing explicit computations for standard spaces, establishing their geometric significance, and identifying conditions for maximal roundness.

Contribution

It introduces the concept of coroundness, relates it to smoothness and convexity, and offers new criteria for maximal roundness greater than one in Banach spaces.

Findings

Computed roundness for $\, ext{ell}^{p}$, $ ext{Lebesgue-Bochner}$, and Schatten spaces

Established relations between roundness, coroundness, and geometric properties

Provided examples of Banach spaces with specific roundness characteristics

Abstract

The maximal roundness of a metric space is a quantity that arose in the study of embeddings and renormings. In the setting of Banach spaces, it was shown by Enflo that roundness takes on a much simpler form. In this paper we provide simple computations of the roundness of many standard Banach spaces, such as $\ell^{p}$, the Lebesgue-Bochner spaces $\ell^{p}(\ell^{q})$ and the Schatten ideals $S_{p}$. We also introduce a property that is dual to that of roundness, which we call coroundness, and make explicit the relation of these properties to the geometric concepts of smoothness and convexity of Banach spaces. Building off the work of Enflo, we are then able to provide multiple non-trivial equivalent conditions for a Banach space to possess maximal roundness greater than $1$. Using these conditions, we are able to conclude that certain Orlicz spaces possess non-trivial values of roundness and coroundness. Finally, we also use these conditions to provide an explicit example of a $2$-dimensional Banach space whose maximal roundness is not equal to that of its dual.