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Points on Hemispheres

Jan Fricke
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Abstract

We will show that for any $n\ge N$ points on the $N$-dimensional sphere $S^N$ there is a closed hemisphere which contains at least $\lfloor\frac{n+N+1}{2}\rfloor$ of these points. This bound is sharp and we will calculate the amount of sets which realize this value. If we change to open hemispheres things will be easier. For any $n$ points on the sphere there is an open hemisphere which contains at least $\lfloor\frac{n+1}{2}\rfloor$ of these points, independent of the dimension. This bound is sharp.

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Jan Fricke. 2006-10-23. Points on Hemispheres. https://arxiv.org/abs/math/0610140

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