On extremums of sums of powered distances to a finite set of points

In this paper we investigate the extremal properties of the sum $$\sum_{i=1}^n|MA_i|^{\lambda},$$ where $A_i$ are vertices of a regular simplex, a cross-polytope (orthoplex) or a cube and $M$ varies on a sphere concentric to the sphere circumscribed around one of the given polytopes. We give full characterization for which points on $\Gamma$ the extremal values of the sum are obtained in terms of $\lambda$. In the case of the regular dodecahedron and icosahedron in $\mathbb{R}^3$ we obtain results for which values of $\lambda$ the corresponding sum is independent of the position of $M$ on $\Gamma$. We use elementary analytic and purely geometric methods.