Roundness properties of Banach spaces

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 l(p), the Lebesgue-Bochner spaces l(p)(l(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 nontrivial 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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper explores the roundness and coroundness of Banach spaces, providing simple computations and explicit examples. Equivalent conditions for maximal roundness greater than 1 are given, and conclusions about non-trivial values in certain Orlicz spaces are drawn. Additionally, an explicit example of a 2-dimensional Banach space with non-equal maximal roundness to its dual is provided.