Large Banach spaces with no infinite equilateral sets

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 nonseparable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of 𝓁 1 ([0, 1]). A wider class of renormings of 𝓁 1 ([0, 1]) which admit no uncountable equilateral sets is also considered. M S C 2 0 2 0 46B20, 03E75, 46B26 (primary)

Banach Spaces in Which Large Subsets of Spheres Concentrate

“…Both of the questions were answered recently in the negative in ZFC in [33] and in [34], [33] respectively and stronger negative solutions were obtained consistently in [35] and [23] . The spaces are of density 2 ω .…”

“…The only known so far examples of Banach spaces whose unit spheres do not admit uncountable (1+)-separated sets or uncountable equilateral sets have densities up to 2 ω ( [33,34]). It is not known at the moment if there are Banach spaces of density in the interval (2 ω , 2 2 ω ] with no uncountable (infinite) equilateral sets.…”

“…In the following part of this section we give examples of Banach spaces which show that in some sense the hypotheses of Propositions 39 and 41 cannot be weakened and their conclusions cannot be made stronger. These are examples of [33], [23] and [34]. To show that they do the job, we need a couple of lemmas.…”

“…The spaces are of density 2 ω . All these examples, including the above ZFC examples enjoy rather new metric phenomena in nonseparable Banach spaces, e.g., some of them admit no uncountable, or even no infinite equilateral sets ( [34]), the unit spheres of some of them are the unions of countably many sets of diameters strictly less than 1 ( [23]) or some of them admit no uncountable Auerbach systems ( [33]; for definition of Auerbach systems see Section 2.1; first such example was obtained under CH in [25]). At the same time other results showed that the above-mentioned stronger consistent irregularity properties of Banach spaces of particular types are consistently impossible: all Banach spaces of the form C(K) admit uncountable equilateral sets under MA+¬CH (Theorem of 5.1 [31]), there are geometric dichotomies for Johnson-Lindenstrauss spaces and for some of their renormings under OCA (Theorem 2 of [23]), the spheres of Banach spaces of Shelah induced by anti-Ramsey colorings still admit uncountable (1+)-separated and equilateral sets under MA+¬CH (Theorem 3 (3) of [35]).…”

“…We would like to thank W. Marciszewski for two reasons. First, for motivating the author to prepare a survey seminar presentation concerning the results of papers [31,33,34,23,35]. This lead to some observations which started the research which resulted in this paper.…”