On minimizing sequences for \(k\)-centres
Let \(P\) be a Borel measure on a separable metric space \((E,d)\), \(k\geq 1\), and \(\varphi\) be an increasing function. The author considers the problem of approximating \(P\) by \(k\)-element subsets \(A\subset E\) by minimization of the function \(W(A)=\int \varphi (d(x,A))P(dx)\). For example, if \(E\) is a separable dual Banach space with the property: \(x_n\rightarrow\) weakly*\(x\) and \(\|x_n\|\rightarrow \|x\|\) imply \(\|x_n-x\|\rightarrow 0\), then every minimizing sequence \(A_n\) has a norm convergence subsequence. The problem arises from the investigation of consistency of empirical \(k\)-centres in mathematical statistics. The paper is based on the author's doctoral dissertation.
scientific article; zbMATH DE number 1891274