In a paper I am reviewing, the authors define the following problem and construct an algorithm. They give no further references and I suspect it has appeared somewhere in the literature before.
Let $P_1,\ldots,P_n$ be sets of points in a space with a metric $d$. Define the distance to a set as $$\operatorname{dist}(x, P_i) = \min\{ d(x,p) \,|\, x \in P_i\}.$$ The problem is to compute $$\min_x \sum_{i=1}^n \operatorname{dist}(x, P_i).$$ What is the name of this problem? Has it been discussed in the literature? What if $d(x,P_i) = ||x - P_i||_2?$