Let $S^k$ denote the unit hypersphere in $\mathbb{R} ^{k+1}$. For a point $a\in S^{k}$ let $H_{\epsilon}(a):= {\{ x | \langle x,a \rangle > \epsilon}\}$. If $\epsilon =0$, then $H_{0}(a)$ is simply the open hemisphere centered at the point $a$.
Gales Theorem states that if $n$ and $k$ are positive integers, then there is a set $V \subset S^k$ with $2n+k$ elements such that $|H_{0}(a) \cap V| \geq n$ for each $a \in S^k$.
I'm interested in extensions of this theorem for "smaller caps" - i.e. $\epsilon >0$. That is, I would like to obtain something of the form:
If $n$ and $k$ are positive integers, and $\epsilon$>0 then there is a set $V \subset S^k$ with $2n+k$ elements such that $|H_{\epsilon}(a) \cap V| \geq f(n,k,\epsilon$ ) for each $a \in S^k$.
I'm primarily interested in the case where $\epsilon$ is small $\approx \Theta \frac{1}{\sqrt{k}}$
Any pointers or references would be greatly appreciated. Upper or lower bounds are also of interest.