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.