Power Invariants of Certain Point Sets

Abstract

Point sets {A₁,...,A_n} in $\mathbb{R}^d$, d ≥ 2, are considered that have barycenter at the origin and, for a certain collection of even exponents 2,4,...,2p, possess “power invariants” I_k in the following sense. Let S^d-1(R) be the sphere with center at the origin and radius R and let M ∈ S^d-1(R). Then the $I_k(R)=\sum_{i=1}^n |MA_i|^2k,k=1,\ldots,p$, do not depend on the position of M on S^d-1(R). Bibliography: 14 titles.