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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.

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  • $\begingroup$ What is $f$ and what is $\Theta(\frac{1}{\sqrt{k}})$? $\endgroup$
    – Fomalhaut
    Commented Feb 19, 2019 at 20:39
  • $\begingroup$ @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. $\endgroup$
    – mm8511
    Commented Feb 21, 2019 at 1:59
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    $\begingroup$ 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 $\vert H_\epsilon(a) \cap V\vert \geq n$. $\endgroup$ Commented Feb 21, 2019 at 2:08

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