# Gale's Theorem Analog for smaller caps

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.

• What is $f$ and what is $\Theta(\frac{1}{\sqrt{k}})$? Commented Feb 19, 2019 at 20:39
• @tomislavostojich My apologies. $f$ is just some arbitrary “bounding function.” I’m looking for a bound in terms of $n$, $k$. And $\epsilon$... $\Theta ( x )$ is notation for the class of functions that “grow like $x$“ - up to constants. Like Big-O notation. Commented Feb 21, 2019 at 1:59
• Gale's Theorem for $n$ large and small $k$ says that among $2n$ points any hemisphere contains at least $n$=approx. half of the points. So fixing $\epsilon > 0$ and letting $p$ be the fraction of the surface of the sphere covered by $H_\epsilon(a)$, you might hope to prove that there are $\approx n/p$ points such that $\vert H_\epsilon(a) \cap V\vert \geq n$. Commented Feb 21, 2019 at 2:08