The tight span of an antipodal metric space: Part II - Geometrical properties

Abstract

Suppose that X is a finite set and let R-x denote the set of functions that map X to R. Given a metric d on X, the tight span of (X, d) is the polyhedral complex T (X, d) that consists of the bounded faces of the polyhedron P(X, d) := {f is an element of R-x : f(x) + f (y) greater than or equal to d(x, y)}. In a previous paper we commenced a study of properties of T(X, d) when d is antipodal, that is, there exists an involution sigma : X --> X: x --> (x) over bar so that d(x, y) + d(y,(x) over bar) = d(x, (x) over bar) holds for all x, y c X. Here we continue our study, considering geometrical properties of the tight span of an antipodal metric space that arise from a metric with which the tight span comes naturally equipped. In particular, we introduce the concept of cell-decomposability for a metric and prove that the tight span of such a metric is the union of cells, each of which is isometric and polytope isomorphic to the tight span of some antipodal metric. In addition, we classify the antipodal cell-decomposable metrics and give a description of the polytopal structure of the tight span of such a metric.