Existence of k-media and medians

Abstract

Abstract The notion of median, or Fermat point, of a finite set, has been recently generalized in two ways (see P.L. Papini and J. Puerto, preprint 2002; E. Alvoni, preprint 2002). Here we study conditions on the underlying space related to the existence of solutions concerning the second generalization. Let X be a real Banach space; consider a finite subset A of X containing n elements and let k be an integer between 2 and n. For x in X, consider the distances among x and k points of A nearest to x; set μ k (A,x)=average of these numbers. We want to minimize μ k (A,x) (for x∈X): a solution of this problem, if it exists, will be called a k-medium of A. The function μ k (A,x) is neither convex (or quasi-convex), nor concave; therefore the existence of solutions does not follow from standard results on convex functions. Here we shall discuss existence of k-media; we will show that if the space X is such that every finite set A has a median (also called a Fermat point), then the same is true for k-media; in particular, in reflexive spaces, as well as in several classical spaces, k-media always exist.