Averaging the k largest distances among n: k-centra in Banach spaces

Abstract

Given a Banach space X let A ⊂ X containing at least k points. In location theory, reliability analysis, and theoretical computer science, it is useful to minimize the sum of distances from the k furthest points of A: this problem has received some attention for X a finite metric space (a network), see, e.g., [Discrete Appl. Math. 109 (2001) 293]; in the case X = En, k = 2 or 3, and A compact some results have been given in [Math. Notes 59 (1996) 507]; also, in the field of theoretical computer science it has been considered in [T. Tokuyama, Minimax parametric optimization problems in multidimensional parametric searching, in: Proc. 33rd Annu. ACM Symp. on Theory of Computing, 2001, pp. 75–84]. Here we study the above problem for a finite set A ⊂ X, generalizing—among others things—the results in [Math. Notes 59 (1996) 507].