Banach spaces in which large subsets of spheres concentrate
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 JohnsonLindenstrauss 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 DOI:
 10.48550/arXiv.2104.05335
 arXiv:
 arXiv:2104.05335
 Bibcode:
 2021arXiv210405335K
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  General Topology;
 Mathematics  Logic
 EPrint:
 Added final section with open problems