In the Heilbronn triangle problem, $n$ points are arranged with restrictions so that the smallest triangle made by a 3 point subset is as large as possible.
Consider the problem on a unit sphere using spherical triangles. For six points, a triangular prism with triangle edge $\sqrt{2}$ and height $1/\sqrt{3}$ seems optimal, giving 14 spherical triangles of area $\pi/2$ and 6 spherical triangles of area $3\pi/2$, shown below as polygons. All vertices are distance 1 from the center.
That looks like a very nice result. Using the Euler-Gauss-Simpsons Theory, somebody else found it first. Extending the problem to more points seems non-trivial, so I'm wondering if the problem has already been studied under another name before I spend much time on it. Are optimal results known for 7+ points?