Given a metric space $(X,d)$ and finite number of points $(x_i)_{i=1}^n$ the Voronoi diagram (or the Dirichlet cell) $C_i$ is given by $$ C_i = \{x\in X:d(x,x_i)<\min\limits_{j\neq i}d(x,x_j)\}. $$
I have two question:
Does anybody know if it was considered the case with different distance function, namely $$ C'_i = \{x\in X:d_i(x,x_i)<\min\limits_{j\neq i}d_j(x,x_j)\}. $$ where $(d_i)_{i=1}^n$ are all metric on $X$.
For the latter case - is it possible in general to find a metric $d'$ on $X$ such that $C'_i$ define above admits the representation $$ C'_i = \{x\in X:d'(x,x_i)<\min\limits_{j\neq i}d'(x,x_j)\}. $$ for any $i$.
Edited: the trivial answer on 2. is to set $d'(x,x) = 0$, $d'(x,y)=1$ if $x,y$ are from one cell and $d'(x,y)=2$ if they are from the different cells. I wonder if 2. can be solved without construction $C'_i$ first. Since the question now is not mathematically correct, I'll put a reference request tag.
