Closed linear spaces consisting of strongly norm attaining Lipschitz mappings

TL;DR

This paper investigates the existence and properties of finite and infinite-dimensional linear subspaces of Lipschitz functions that strongly attain their norm, providing comprehensive results for various types of metric spaces.

Contribution

It establishes that infinite metric spaces always admit such subspaces and characterizes their sizes and structures, especially for $\sigma$-precompact and spaces containing [0,1].

Findings

Infinite metric spaces always have finite-dimensional subspaces of strongly norm-attaining Lipschitz functions.

For $\sigma$-precompact spaces, these subspaces are always separable and isomorphically polyhedral.

Spaces containing [0,1] can have infinite-dimensional such subspaces.

Abstract

Given a pointed metric space $M$, we study when there exist $n$-dimensional linear subspaces of $\operatorname{Lip}_0(M)$ consisting of strongly norm-attaining Lipschitz functionals, for $n\in\mathbb{N}$. We show that this is always the case for infinite metric spaces, providing a definitive answer to the question. We also study the possible sizes of such infinite-dimensional closed linear subspaces $Y$, as well as the inverse question, that is, the possible sizes of the metric space $M$ given that such a subspace $Y$ exists. We also show that if the metric space $M$ is $\sigma$-precompact, then the aforementioned subspaces $Y$ need to be always separable and isomorphically polyhedral, and we show that for spaces containing $[0,1]$ isometrically, they can be infinite-dimensional.