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2003 The Hypermetric Cone on Seven Vertices
Michel Deza, Mathieu Dutour
Experiment. Math. 12(4): 433-440 (2003).

Abstract

The hypermetric cone {\small $HYP_n$} is the set of vectors {\small $(d_{ij})_{1\leq i< j\leq n}$} satisfying the inequalities

\sum_{1\leq i<j\leq n} b_ib_jd_{ij}\leq 0\mbox{with}b_i\in\Z\mbox{and}\sum_{i=1}^{n}b_i=1\,.

A Delaunay polytope of a lattice is called extreme if the only affine bijective transformations of it into a Delaunay polytope, are the homotheties; there is correspondence between such Delaunay polytopes and extreme rays of {\small $HYP_n$}. We show that unique Delaunay polytopes of root lattices {\small $A_1$} and {\small $E_6$} are the only extreme Delaunay polytopes of dimension at most 6. We describe also the skeletons and adjacency properties of {\small $HYP_7$} and of its dual.

The computational technique used is polyhedral, i.e., enumeration of extreme rays, using the program cdd [Fukuda 03], and groups to reduce the size of the computations.

Citation

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Michel Deza. Mathieu Dutour. "The Hypermetric Cone on Seven Vertices." Experiment. Math. 12 (4) 433 - 440, 2003.

Information

Published: 2003
First available in Project Euclid: 18 June 2004

zbMATH: 1101.11021
MathSciNet: MR2043993

Subjects:
Primary: 11H06
Secondary: 52B20

Keywords: cone , Delaunay polytope , hypermetric , lattice

Rights: Copyright © 2003 A K Peters, Ltd.

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Vol.12 • No. 4 • 2003
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