On minimizing sequences for k-centres

Let P be a Borel measure on a separable metric space (E, d). Given an integer k greater than or equal to 1 and a nondecreasing function phi: R+--> R+ we seek to approximate P by a subset of E which, amongst all subsets of at most k elements, minimizes the function W-k (A, P) := integralphi(d(x,A))P(dx). Any set that minimizes W-k(.,P) is called a k-centre of P. We study the convergence of W-k(., P)-minimizing sequences in noncompact spaces. As an application we prove a consistency result for empirical k-centres. (C) 2002 Elsevier Science (USA). All rights reserved.