Finding N most distinct values in set given distance metric
This problem generalizes maximum independent set and its relatives. Indeed, suppose that you could solve your problem. Then you could use the solution for deciding whether a graph contains an independent set of size $k$. Given a graph $G = (V,E)$, form a metric space by setting $d(u,v) = 1$ if $\{u,v\} \in E$ and $d(u,v) = 2$ if $\{u,v\} \notin E$. The graph contains an independent set of size $k$ iff there are $k$ points in the metric space in which the sum of distances is $k(k-1)$.