Asymptotic of an optimal location problem (Q1851453)

It is known that for a uniform demand density on the unit \(d\)-cube the minimum mean distance to the closest of \(n\) points, asymptotically equals \(C_dn^{-1/d}\). In particular \(C_2\) is known to be the average distance to the center for a uniform hexagon of unit area. In this paper the asymptotic result is extended to non uniform unit density \(f\), yielding the same value multiplied by \((\int f^p(x)\,dx)^{1/p}\) with \(p=d/(d+1)\).