Asymptotic optimal location of facilities in a competition between population and industries

Abstract

We consider the problem of optimally locating a given number $k$ of points in ${\mathbb R}^n$ for an integral cost function which takes into account two measures $\varphi^+$ and $\varphi^-$. The points represent for example new industrial facilities that have to be located, the measure $\varphi^+$ representing in this case already existing industries that want to be close to the new ones, and $\varphi^-$ representing private citizens who want to stay far away. The asymptotic analysis as $k\to\infty$ is performed, providing the asymptotic density of optimal locations.