On minimizing sequences for k-centres Original Research Article

Let P be a Borel measure on a separable metric space (E,d). Given an integer k⩾1 and a nondecreasing function φ : R+→R+ we seek to approximate P by a subset of E which, amongst all subsets of at most k elements, minimizes the function Wk(A,P)≔∫φ(d(x,A))P(dx). Any set that minimizes Wk(·,P) is called a k-centre of P. We study the convergence of Wk(·,P)-minimizing sequences in noncompact spaces. As an application we prove a consistency result for empirical k-centres.