Symmetrically separated sequences in the unit sphere of a Banach space

TL;DR

This paper proves a symmetric version of Kottman's theorem, showing the existence of large subsets in the unit sphere of Banach spaces with pairwise sum and difference norms exceeding 1, and provides quantitative estimates under structural conditions.

Contribution

It establishes a symmetric variant of Kottman's theorem for Banach spaces, including quantitative bounds and renorming results based on structural properties.

Findings

Existence of infinite subsets with pairwise sum/difference norms > 1

Quantitative estimates for ε under structural conditions

Renorming results for Banach spaces

Abstract

We prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite-dimensional Banach space contains an infinite subset $A$ with the property that $\|x\pm y\| > 1$ for distinct elements $x,y\in A$, thereby answering a question of J. M. F. Castillo. In the case where $X$ contains an infinite-dimensional separable dual space or an unconditional basic sequence, the set $A$ may be chosen in a way that $\|x\pm y\| \geqslant 1+\varepsilon$ for some $\varepsilon > 0$ and distinct $x,y\in A$. Under additional structural properties of $X$, such as non-trivial cotype, we obtain quantitative estimates for the said $\varepsilon$. Certain renorming results are also presented.