On minimizing sequences for k-centres

Authors:
Jüri Lember
Affiliations:
EURANDOM, P.O. Box 513-5600 MB Eindhoven, Netherlands
Venue:
Journal of Approximation Theory
Year:
2003

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Abstract

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.