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

TL;DR

This paper studies the extremal values of sums of powered distances from a point on a sphere to vertices of regular polytopes, providing characterizations and conditions for independence of the sum from the point’s position.

Contribution

It offers a complete characterization of extremal points for sums of powered distances to vertices of regular polytopes on concentric spheres, including special cases for dodecahedron and icosahedron.

Findings

Identifies points on the sphere where the sum attains extremal values.

Determines conditions under which the sum is independent of the point's position.

Provides geometric and analytic methods for these characterizations.

Abstract

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.