Abstract
Suppose that X is a finite set and let ℝX 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 ∈ ℝX : f(x)+f(y) ≥ 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 σ: X→ X: x↦ $\overline{x}$ so that d(x,y)+d(y,$\overline{x}$)=d(x,$\overline{x}$) holds for all x,y ∈ 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.
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Huber, K., Koolen, J. & Moulton, V. The Tight Span of an Antipodal Metric Space: Part II—Geometrical Properties. Discrete Comput Geom 31, 567–586 (2004). https://doi.org/10.1007/s00454-004-0777-3
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DOI: https://doi.org/10.1007/s00454-004-0777-3