The tight span of an antipodal metric space. I: combinatorial properties (Q2576852)

For a finite metric space \((X,d)\) the tight span \(T(X,d)\) of \((X, d)\) is the polyhedral complex that consists of the bounded faces of the polyhedron \(P(X,d) = \{f\in\mathbb R^X\mid f(x) + f(y)\geq d(x,y)\) for all \(x,y\in X\}\). It was introduced by \textit{J. R. Isbell} in [Comment. Math. Helv. 39, 65--76 (1964; Zbl 0151.30205)] and subsequently rediscovered by \textit{A. Dress} in [Adv. Math. 53, 321--402 (1984; Zbl 0562.54041)]. In Part II of this paper [Discrete Comput. Geom. 31, No.~4, 567--586 (2004; Zbl 1081.52015)] the authors have studied geometric properties of the tight span \(T(X,d)\) of an antipodal metric space \((X,d)\), i.e. of a finite metric space \((X,d)\) which has an involution \(\sigma : X \to X\) such that, for all \(x,y\in X\), \(d(x,y)+d(y,\sigma(x)) = d(x, \sigma(x))\). In Part I they investigate combinatorial properties of the tight span of a finite metric space. In particular, they show that a finite metric space \((X,d)\) with cardinality \(\geq 2\) is antipodal if and only if its tight span \(T(X,d)\) contains a unique maximal cell (and is therefore a convex polytope). In this case there exists a natural bijection between the facets of \(T(X,d)\) and the edges of the so-called underlying graph of \((X,d)\). Moreover, it is shown that every antipodal metric space with exactly 6 points is totally split-decomposable in the sense of \textit{H.-J. Bandelt} and \textit{A. W. M. Dress} [Adv. Math. 92, No.~1, 47--105 (1992; Zbl 0789.54036)].