Given a finite set A={a 1 ,a 2 ,...,a n } in a normed linear space X; for x X, let π i (x) be a permutation of {1,2,...,n} such that ||x-a π 1 ( x ) ||=<||x-a π 2 ( x ) ||=<...=<||x-a π n ( x ) ||. We consider the following problem: for 1=<k=<n, let 1k i = 1 k ||x-a π i ( x ) || be the average distance to the k nearest points from a point x of the space; we are interested in minimizing this average when x describes the space X and in finding optimal solutions. This problem, which has a clear practical meaning, seems to have received little attention. Several properties of the solutions are proved.