Antipodal sets in infinite dimensional Banach spaces

The following strengthening of the Elton-Odell theorem on the existence of a $(1+ϵ)-$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 $x\mathrm{\ne }y$ there exists $f\in B_{X^{\ast }}$ such that $d\leq f(x)-f(y)$and$f(y)\leq f(z)\leq f(x)$,forall$z\in S$.