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 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.

Large Banach spaces with no infinite equilateral sets

“…However, that is a renorming of a space $$C_0(K)$$ for $$K$$ locally compact and scattered, so it is $$c_0$$‐saturated (by [15]). Since a result in [13] says that any Banach space which contains an isomorphic copy of $$c_0$$ admits an infinite equilateral set, we conclude that spaces of [12] admit such infinite sets.…”

“…In particular, this solves the question of whether there is a nonseparable Banach space with no uncountable equilateral set ([11, 14], Problem 293 of [7]). Another absolute construction of a nonseparable Banach space with no uncountable equilateral set is being presented at the same time in a paper by the first author [12]. However, that is a renorming of a space $$C_0(K)$$ for $$K$$ locally compact and scattered, so it is $$c_0$$‐saturated (by [15]).…”