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 ${\phi }^{+}$ and ${\phi }^{-}$. The points represent for example new industrial facilities that have to be located, the measure ${\phi }^{+}$ representing in this case already existing industries that want to be close to the new ones, and ${\phi }^{-}$ 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.

25 pages, 3 figures, The final publication is available at http://annaliscienze.sns.it