Abstract :
Given a finite set A={a1,a2,…,an} in a normed linear space X; for x∈X, let πi(x) be a permutation of {1,2,…,n} such that ||x−aπ1(x)||⩽||x−aπ2(x)||⩽⋯⩽||x−aπn(x)||. We consider the following problem: for 1⩽k⩽n, let 1k∑i=1k||x−aπi(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.
