Antipodal sets in infinite dimensional Banach spaces

TL;DR

This paper strengthens the Elton-Odell theorem by demonstrating the existence of an infinite antipodal subset in the unit sphere of any infinite dimensional Banach space, with specific separation and ordering properties.

Contribution

It introduces a new form of antipodal set with ordered separation in infinite dimensional Banach spaces, extending previous results on separated sequences.

Findings

Existence of an infinite antipodal subset in the unit sphere

The subset has a uniform separation constant greater than 1

Ordered separation property holds for all pairs in the subset

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 $S\subseteq S_X$ 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 $z\in S$.