On the tight span of an antipodal graph
The tight span of a finite metric space is essentially the `smallest' path geodesic space into which the metric space embeds isometrically [compare \textit{A. W. M. Dress}, Adv. Math. 53, 321-402 (1984; Zbl 0562.54041)]. In this situation, the tight span is also contractible and has a natural cell structure, so that it lends itself naturally to the study of the Cayley graph of a group. As a first step in this study, the tight span of a metric space which arises from the graph metric of an antipodal graph is considered. In particular, some techniques for the study of the tight span of a graph are developed, which are then applied to antipodal graphs. In this way, the polytopal structure of the tight span for special examples of antipodal graphs (e.g., planar antipodal graphs) is found. It is shown that the tight span of the simplest kind of antipodal graphs, the polygon with \(2k\) edges, is the \(k\)-dimensional hypercube. Finally, with the aid of programs written in Mathematica, computer generated examples of tight spans which were made possible by the techniques developed in this paper are presented for 3- and 4-cube, octahedron, icosahedron, dodecahedron and permutahedron with 24 vertices.
scientific article; zbMATH DE number 1453322