Large Banach spaces with no infinite equilateral sets

TL;DR

This paper constructs non-separable, strictly convex Banach spaces of density continuum that lack infinite equilateral sets, expanding understanding of geometric properties in Banach space theory.

Contribution

It introduces new renormings of ([0,1]) that do not contain uncountable equilateral sets, demonstrating novel geometric configurations.

Findings

Existence of non-separable Banach spaces with no infinite equilateral sets

Strictly convex renormings of ([0,1]) without uncountable equilateral sets

Broader class of renormings with similar properties

Abstract

A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist non-separable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of $\ell_1([0,1])$. A wider class of renormings of $\ell_1([0,1])$ which admit no uncountable equilateral sets is also considered.