Cut points in metric spaces

Abstract

In this note, we will define topological and virtual cut points of finite metric spaces and show that, though their definitions seem to look rather distinct, they actually coincide. More specifically, let X denote a finite set, and let D : X x X -> R : (x, y) -> xy denote a metric defined on X. The tight span T(D) of D consists of all maps f is an element of R-X for which f (x) = sup(y is an element of X) (xy - f (x)) holds for all x is an element of X. Define a map f is an element of T(D) to be a topological cut point of D if T(D) - {f} is disconnected, and define it to be a virtual cut point of D if there exists a bipartition (or split) of the support supp(f) of f into two non-empty sets A and B such that ab = f(a) + f(b) holds for all points a is an element of A and b is an element of B. It will be shown that, for any given metric D, topological and virtual cut points actually coincide, i.e., a map f is an element of T(D) is a topological Cut point of D if and only if it is a virtual cut point of D. (C) 2007 Elsevier Ltd. All rights reserved.