Banach spaces in which large subsets of spheres concentrate

TL;DR

This paper constructs a nonseparable Banach space where uncountable subsets of the sphere are highly concentrated, providing a negative answer to a nonseparable version of Kottman's theorem and revealing new geometric properties.

Contribution

It introduces a novel nonseparable Banach space with specific concentration properties, addressing a central open problem in the geometry of Banach spaces.

Findings

Uncountable subsets of the sphere contain points less than 1 apart.

The space contains no uncountable Auerbach system.

The space contains no uncountable equilateral set.

Abstract

We construct a nonseparable Banach space $\mathcal X$ (actually, of density continuum) such that any uncountable subset $\mathcal Y$ of the unit sphere of $\mathcal X$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon$ for some $\varepsilon>0$). This solves in the negative the central problem of the search for a nonseparable version of Kottman's theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains neither an uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson-Lindenstrauss space induced by an $\mathbb R$-embeddable almost disjoint family of subsets of $\mathbb N$. We also show that this special feature of the almost disjoint family is essential to obtain the above properties.