Abstract
In this chapter we establish the fundamental connection existing between hyper-metric spaces and Delaunay polytopes. Hence, for a hypermetric distance space (X, d), one may speak of its associated Delaunay polytope P d ; the case when (X, d) is ℓ 1-embeddable corresponding to the case when P d can be embedded in a parallepiped. As an application of this connection, one can show polyhedrality of the hypermetric cone; several proofs for this fact are given in Section 14.2. As another application (and using the classification of the irreducible root lattices), one can characterize the graphs whose shortest path metric is hypermetric or ℓ l-embeddable. Such graphs arise essentially from cocktail-party graphs, half-cube graphs, and a single graph on 56 nodes (the Gosset graph) by taking Cartesian products and isometric subgraphs (see Section 14.3). We group in Section 14.4 several results concerning spherical representations of distance spaces and the radius of Delaunay polytopes.
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© 1997 Springer-Verlag Berlin Heidelberg
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Deza, M.M., Laurent, M. (1997). Hypermetrics and Delaunay Polytopes. In: Geometry of Cuts and Metrics. Algorithms and Combinatorics, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04295-9_14
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DOI: https://doi.org/10.1007/978-3-642-04295-9_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04294-2
Online ISBN: 978-3-642-04295-9
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