Symmetrically separated sequences in the unit sphere of a Banach space

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 $\Vert x\pm y\Vert >1$ 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 $\Vert x\pm y\Vert ⩾1+\epsilon$ for some $\epsilon >0$ and distinct $x,y\in A$. Under additional structural properties of $X$, such as non-trivial cotype, we obtain quantitative estimates for the said $\epsilon$. Certain renorming results are also presented.

19 pp

References in corpus (2)

• The geometry of Minkowski spaces -- a survey. Part I
• Uncountable sets of unit vectors that are separated by more than 1

Cited by in corpus (2)

• Separated sets and Auerbach systems in Banach spaces
• A note on symmetric separation in Banach spaces