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The tight span of an antipodal metric space: Part II - Geometrical properties
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Faculty of Science and Technology, Biology, The Linnaeus Centre for Bioinformatics.
2004 (English)In: Discrete & Computational Geometry, ISSN 0179-5376, Vol. 31, no 4, 567-586 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2004. Vol. 31, no 4, 567-586 p.
Keyword [en]
splitstree
Identifiers
URN: urn:nbn:se:uu:diva-71815OAI: oai:DiVA.org:uu-71815DiVA: diva2:99726
Available from: 2005-05-12 Created: 2005-05-12 Last updated: 2011-01-12

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