Equilateral dimension of some classes of normed spaces

TL;DR

This paper investigates the maximum number of equidistant points in certain finite-dimensional normed spaces, confirming the conjecture in specific classes and providing bounds for spaces close to these classes.

Contribution

It proves the conjecture for permutation-invariant, Orlicz-Musielak, and certain subspaces of  spaces, and extends bounds to spaces close to these classes.

Findings

Confirmed the conjecture for permutation-invariant spaces.

Established the conjecture for Orlicz-Musielak spaces.

Provided bounds for spaces close to the studied classes.

Abstract

An equilateral dimension of a normed space is a maximal number of pairwise equidistant points of this space. The aim of this paper is to study the equilateral dimension of certain classes of finite dimensional normed spaces. The well-known conjecture states that the equilateral dimension of any $n$-dimensional normed space is not less than $n+1$. By using an elementary continuity argument, we establish it in the following classes of spaces: permutation-invariant spaces, Orlicz-Musielak spaces and in one codimensional subspaces of $\ell^n_{\infty}$. For smooth and symmetric spaces, Orlicz-Musielak spaces satisfying an additional condition and every $(n-1)$-dimensional subspace of $\ell^{n}_{\infty}$ we also provide some weaker bounds on the equilateral dimension for every space which is sufficiently close to one of these. This generalizes the result of Swanepoel and Villa concerning the $\ell_p^n$ spaces.