On isometric embeddings into the set of strongly norm-attaining Lipschitz functions

TL;DR

This paper investigates the structure of strongly norm-attaining Lipschitz functions on metric spaces, showing conditions under which they contain or do not contain isometric copies of c_0, thus answering a previously open question.

Contribution

It constructs an example of an infinite metric space where SNA(M) lacks an isometric c_0, and proves that SNA(M) contains c_0 when M is not uniformly discrete, resolving an open problem.

Findings

SNA(M) does not always contain an isometric c_0.

SNA(M) contains c_0 if M is not uniformly discrete.

Results extend to certain non-separable spaces.

Abstract

In this paper, we provide an infinite metric space $M$ such that the set $\mbox{SNA}(M)$ of strongly norm-attaining Lipschitz functions does not contain a subspace which is isometric to $c_0$. This answers a question posed by Antonio Avil\'es, Gonzalo Mart\'inez Cervantes, Abraham Rueda Zoca, and Pedro Tradacete. On the other hand, we prove that $\mbox{SNA}(M)$ contains an isometric copy of $c_0$ whenever $M$ is a metric space which is not uniformly discrete. In particular, the latter holds true for infinite compact metric spaces while it does not for proper metric spaces. Some positive results in the non-separable setting are also given.