On the Basic Properties and the Structure of Power Cells

Abstract

Given a set T⊆Rn and a nonnegative function r defined on T, we consider the power of x∈Rn with respect to the sphere with center t∈T and radius rt, that is, prx,t:=x-t2-r2t, with · denoting the Euclidean distance. The corresponding power cell of s∈T is the set (Formula presented.) We study the structure of such cells and investigate the assumptions on r that allow for generalizing known results on classical Voronoi cells.