I am working on the following problem: I want to tile a space (in the following assume a two-dimensional Euclidian plane). I want to do this in a similar fashion as Voronoi diagrags do, however, I do not want to consider single points but instead groups of single points.
Regular Voronoi diagrams consider a set $S$ where $p \in S$ are points in $\mathbb{R}^2$.
The Voronoi region $VR(p,S)$ is then defined by
$VR(p,S) = \bigcap_{{q \in S\setminus\{p\}}} D(p,q)$ where
$D(p,q)=\{x \in \mathbb{R}^2: |p-x|<|q-x|\}$.
I am interested in a somewhat similar concept. However, my set $S$ is a set of sets such that $P \in S$ is itself a set where $p \in P$ are points in $\mathbb{R}^2$.
The region of interest $UR(P,S)$ is now defined by
$UR(P,S) = \bigcap_{{Q \in S\setminus\{P\}}} D(P,Q)$ where
$D(P,Q)=\{x \in \mathbb{R}^2: |p_1-x|+...+|p_n-x|<|q_1-x|+...+|q_m-x|, \forall p_i \in P, q_i \in Q\}$.
What line of reseach is studying the concept defined by $UR(P,S)$? What is the name of this concept in the literature? Thank you a lot!