Spherical triangular configurations with invariant geometric mean

The main objective is to characterize all configurations of three distinct points on the n-dimensional sphere that have the same Riemannian geometric mean and find efficient ways to compute such invariant. The regular case, when the points form the vertices of an equilateral spherical triangle, appears as the global minimum of an appropriate cost function. As a warm-up, and also to get more insight for the spherical case, we first develop our ideas for configurations in the Euclidean space Rn. In both cases, the theoretical results are supported by numerical experiments and illustrated by meaningful plots.