Given a finite set A={a(1),a(2),...,a(n)} in a normed linear space X; for x is an element of X, let pi(i)(x) be a permutation of {1,2,...,n} such that parallel tox-a(pi1(x))parallel to less than or equal to parallel tox-a(pi2(x))parallel to less than or equal to ... less than or equal to parallel tox-a(pin(x))parallel to. We consider the following problem: for 1 less than or equal to k less than or equal to n, let 1/k Sigma(i=1)(k) parallel tox-a(pii(x))parallel to 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. (C) 2003 Elsevier B.V. All rights reserved.
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