This is an announcement for the paper “Antipodal sets in infinite dimensional Banach spaces” by Eftychios Glakousakishttps://arxiv.org/find/math/1/au:+Glakousakis_E/0/1/0/all/0/1, Sophocles Mercourakishttps://arxiv.org/find/math/1/au:+Mercourakis_S/0/1/0/all/0/1.
Abstract: The following strengthening of the Elton-Odell theorem on the existence of a $(1+\epsilon)$−separated sequences in the unit sphere $S_X$ of an infinite dimensional Banach space $X$ is proved: There exists an infinite subset $S\subset S_X$ and a constant$d>1$, satisfying the property that for every $x, y\in S$ with $x\neq y$ there exists $f\in B_{X^*}$ such that $d\leq f(x)-f(y)$ and $f(y)\leq f(z)\leq f(x)$, for all $z\in S$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.02002