# Optimal way to place a given number of points in a region?

Let $A\subset\mathbb{R}²$ and $n\in\mathbb{N}$ be a given natural number. How to find a finite subset of $A$, $P=${$p_1,...,p_n$} such that $\int_A f_P(x)$ is minimum, where $f_P(x) = min${$d(x,p_1),...,d(x,p_n)$} and $d$ is the usual distance? Supose a certain number of hospitals must be placed in a city. How should them be placed so that on average the length from any point in the city to the nearest hospital is the less as possible?

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What are your thoughts? – leonbloy Dec 20 '12 at 16:14
If the usual distance is used, the resulting function $f_P$ is composed of 'parts' of surfaces of cones. – Sgernesto Dec 20 '12 at 17:35
This sounds non trivial. You might want to read about Centroidal Voronoi Tessellations (relate, though not quite the same thing -minimizes average square distance rather than average distance) and perhaps k-medians clustering (for samples instead of a distribution). – leonbloy Dec 20 '12 at 21:04