On the Basic Properties and the Structure of Power Cells

Given a set T⊆ Rⁿ T ⊆ R n and a nonnegative function r defined on T , we consider the power of x∈ Rⁿ x ∈ R n with respect to the sphere with center t∈ T t ∈ T and radius r( t) , r t , that is, pᵣ( x,t) :=\| x-t\| ²-r²( t) , p r x , t : = x - t 2 - r 2 t , with \| · \| · denoting the Euclidean distance. The corresponding power cell of s∈ T s ∈ T is the set aligned CTʳ(s):=∈ Rⁿ: pᵣ(x,s)≤ pᵣ(x,t),\ for all \ t∈ T\. aligned C T r ( s ) : = { x ∈ R n : p r ( x , s ) ≤ p r ( x , t ) , for all t ∈ T } . We study the structure of such cells and investigate the assumptions on r that allow for generalizing known results on classical Voronoi cells.