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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:

  1. 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 $X$.

  2. 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.

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up vote 2 down vote accepted

You might examine this classic (1985) paper by Paul Chew: "Voronoi diagrams based on convex distance functions." Google Scholar says this has subsequently been cited by 190(!) papers. Another key search term is "generalized Voronoi diagrams," e.g., "Fast computation of generalized Voronoi diagrams using graphics hardware." But I am quite uncertain if these are relevant to your questions, which I have to admit are not too clear to me...

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Thank you, Joseph. – Ilya Oct 3 '11 at 8:22

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