This is an announcement for the paper “Symmetrically separated sequences in the unit sphere of a Banach space” by Petr Hájekhttps://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1, Tomasz Kaniahttps://arxiv.org/find/math/1/au:+Kania_T/0/1/0/all/0/1, Tommaso Russohttps://arxiv.org/find/math/1/au:+Russo_T/0/1/0/all/0/1.

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 unconditional basic sequence, the set $A$ may be chosen in a way that $|x\pm y|>1+\epsilon$ 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.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1711.05149