# A questions concerning Laguerre/Voronoi tessellations

Fix $$n>1$$ distinguished points $$x_1,\ldots, x_n\in \mathbb R^d$$, the Voronoi tessellations are the subsets $$V_1,\ldots V_n\subset\mathbb R^d$$ defined by

$$V_k~~ := ~~ \big\{x\in\mathbb R^d:\quad |x-x_k|^2~\le ~ |x-x_i|^2,~ \mbox{ for all } i=1,\ldots, n\big\}.$$

Next we introduce $$n$$ translation parameters $$c_1,\ldots, c_n\in (-\infty,\infty)$$, then the Laguerre tessellations are given by, with $$C:=(c_1,\ldots, c_n)\in\mathbb R^n$$,

$$L_k(C)~~ := ~~ \big\{x\in\mathbb R^d:\quad |x-x_k|^2-c_k~\le ~ |x-x_i|^2-c_i,~ \mbox{ for all } i=1,\ldots, n\big\}.$$

Assume we are given a smooth density function $$\rho:\mathbb R^d\to\mathbb R_+$$, and discrete probability weights $$\alpha_1,\ldots,\alpha_n\subset (0,1)$$ satisfying $$\sum_{i=1}^k\alpha_i=1$$, my question is how to find $$C\in\mathbb R^n$$ s.t.

$$\int_{L_i(C)}\rho(x)dx=\alpha_i,~ \mbox{ for all } i=1,\ldots, n.$$

The problem arises in my research, and I am interested in the computation of $$\nabla_{x_i}c_k$$. I have tried with some particular cases, i.e. $$d=1$$ or $$n=2$$, where some concrete analysis can be carried out. But for general $$d$$ and $$n$$, I have no idea whether there is some related literature. Any solution or reference will highly appreciated!