How to select a subset of points from a universal to minimize the distance from outside to inside?

Question

Here is the detailed problem.

I have a set of N points in K-dimension space, called U, and I want select M points of them, called S. For each point p in U, we define the distance from p to S as $$ d(p, S) = \min_{{p_i} \in S} {d(p, p_i) } $$

the target is finding a set S to get

$$ \min_{S} { \max_i {d(p_i, S)}} $$

I believe it's a NP-hard problem (though I fail to prove) and hope to find an approximate algorithm.

Answer 1

That's the Metric $k$-center problem.