2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/117825We 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.Metric GeometryProbability51F99 ;60D05text