Maximal equilateral sets

TL;DR

This paper investigates the size of maximal equilateral sets in various normed spaces, providing bounds, constructions, and conjectures about their maximum possible size relative to the space's dimension.

Contribution

The authors generalize Petty's construction, establish bounds for $m(X)$ in $ ext{l}_p$ spaces, and prove upper bounds for spaces close to $ ext{l}_p^d$, culminating in a conjecture relating $m(X)$ to the dimension.

Findings

Petty's construction extends to certain spaces with $m(X)=4$.

For $1 \\leq p < 2$, $m(\\ell_p(\\Gamma))$ is finite and bounded.

In spaces close to $\\ell_p^d$, $m(X) \\leq d+1$.

Abstract

A subset of a normed space $X$ is called equilateral if the distance between any two points is the same. Let $m(X)$ be the smallest possible size of an equilateral subset of $X$ maximal with respect to inclusion. We first observe that Petty's construction of a $d$-dimensional $X$ of any finite dimension $d\geq 4$ with $m(X)=4$ can be generalised to give $m(X\oplus_1\mathbb{R})=4$ for any $X$ of dimension at least 2 which has a smooth point on its unit sphere. By a construction involving Hadamard matrices we then show that for any set $\Gamma$, $m(\ell_p(\Gamma))$ is finite and bounded above by a function of $p$, for all $1\leq p<2$. Also, for all $p\in[1,\infty)$ and $d\in\mathbb{N}$ there exists $c=c(p,d)>1$ such that $m(X)\leq d+1$ for all $d$-dimensional $X$ with Banach-Mazur distance less than $c$ from $\ell_p^d$. Using Brouwer's fixed-point theorem we show that $m(X)\leq d+1$ for all $d$-dimensional $X$ with Banach-Mazur distance less than 3/2 from $\ell_\infty^d$. A graph-theoretical argument furthermore shows that $m(\ell_\infty^d)=d+1$. The above results lead us to conjecture that $m(X)\leq 1+\dim X$ for all finite-dimensional normed spaces $X$.