An Extension of the Fermat-Torricelli Problem

The Fermat-Torricelli problem is an optimization problem associated with a finite subset $\{a_{j}\}_{j=1}^{q}$ of ℝN and a family $\{c_{j}\}_{j=1}^{q}$ of positive weights. The function F to be minimized is defined by $F(x)=\sum _{j=1}^{q}c_{j}\Vert x-a_{j}\Vert$. In this paper, we extend this problem to the case of volumes.