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

Abstract

In this paper we investigate the extremal properties of the sum $$\sum _{i=1}^n|M{A}_i{|}^{\lambda },$$ 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.

arXiv commentGeom. Dedicata (to appear)

References in corpus (2)

• Polarization optimality of equally spaced points on the circle for discrete potentials
• Riesz Polarization Inequalities in Higher Dimensions