Large Banach spaces with no infinite equilateral sets
Piotr Koszmider, Hugh Wark

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 . A wider class of renormings of which admit no uncountable equilateral sets is also considered.
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Taxonomy
TopicsAdvanced Banach Space Theory · Optimization and Variational Analysis · Approximation Theory and Sequence Spaces
