On the Basic Properties and the Structure of Power Cells

Given a set \(T\subseteq {\mathbb {R}}^{n}\) $T\subseteq {\mathbb{R}}^n$ and a nonnegative function r defined on T,  we consider the power of \(x\in {\mathbb {R}}^{n}\) $x\in {\mathbb{R}}^n$ with respect to the sphere with center \(t\in T\) $t\in T$ and radius \(r\left( t\right) ,\) $r\left(t\right),$ that is, \( {p_r\left( x,t\right) }:=\left\| x-t\right\| ^{2}-r^{2}\left( t\right) ,\) $p_r\left(x,,,t\right):={∥x,-,t∥}^2-r^2\left(t\right),$ with \(\left\| \cdot \right\| \) $∥·∥$ denoting the Euclidean distance. The corresponding power cell of \(s\in T\) $s\in T$ is the set \(\begin{aligned} C_{T}^{r}(s):=\{x\in {\mathbb {R}}^{n}:{ p_r}(x,s)\le {p_r}(x,t),\ \text{ for } \text{ all }\ t\in T\}. \end{aligned}\) $$\begin{array}{c}\hfill C_T^r(s):=\{x\in {\mathbb{R}}^n:p_r(x,s)\le p_r(x,t),\text{for}\text{all}t\in T\}.\end{array}$$ We study the structure of such cells and investigate the assumptions on r that allow for generalizing known results on classical Voronoi cells.