I am trying to find an approach to the next problem.
Let $p(x)$ be a probability density function on $[0,1]^n$, nonzero at all points. For a fixed $n$ find a set of points $\{x_1, x_2, ..., x_n\} \in [0,1]^n$ such that minimizes the functional $$L(x) = \int_{[0,1]^n} \min_{x_k}||x-x_k||p(x)dx.$$
The integral can be decomposed into sum of integrals over Voronoy cells related to the set $\{x_1, x_2, ..., x_n\}$. So this is a kind of partition problem. However I couldnt find any related research. I believe it can be formulated as a problem of equal volume partition or least perimeter partition but can't prove it. Maybe it can be formulated as a physical problem (for example $n$ soap bubbles in a box placement). Please provide keys or links connected to this problem if you know any. Thanks.