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The hypermetric cone on seven vertices. (English) Zbl 1101.11021

For an \(n\)-lattice \(\Lambda\) a sphere with radius \(r\) and centre \(c\) is called empty if its interior does not contain any lattice point, but it contains \(n+1\)-affinely independent points in the boundary. The convex hull of these lattice points is called a Delaunay-polytope (or \(\Lambda\)-polytope). The square of the Euclidean distance between the vertices of such a Delaunay-polytope is a hypermetric, where, in general, a vector \((d_{ij})_{1\leq i<j\leq n}\) is called an \(n\)-hypermetric if \(\sum_{1\leq i<j\leq n} b_ib_j\,d_{ij}\leq 0\) for \(b_1,\dots,b_n\in\mathbb{Z}\) and \(\sum b_i =1\). The set of all \(n\)-hypermetrics form the so called hypermetric cone denoted \(\text{{HYP}}_n\).
Now a Delaunay-polytope is said to be extreme if the only (up to orthogonal transformations and translations) affine bijective transformations of the space which maps Delaunay polytopes onto Delaunay polytopes are the homotheties. It is well known that there is a natural correspondence between hypermetrics induced by extreme Delaunay polytopes and extreme rays of the hypermetric cone.
Using sophisticated computational techniques based on polyhedral theory the authors are able to describe the skeleton and adjacency properties of \(\text{{HYP}}_7\) and thereby they can prove that the only extreme Delaunay polytopes of dimension at most six are the \(1\)-simplex and the Schläfli polytope, which is the unique Delaunay polytope of \(E_6\).
For more details on hypermetrics we refer to the book of M. Deza and M. Laurent, Geometry of Cuts and Metrics, Berlin, Springer (1997; Zbl 0885.52001).

MSC:

11H06 Lattices and convex bodies (number-theoretic aspects)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

Citations:

Zbl 0885.52001

Software:

cdd; nauty
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References:

[1] Assouad P., Compte Rendus de I’Academie des Sciences de Paris 294 pp 439– (1982)
[2] Baranovskii E. P., Mathematical Notes 10 pp 827– (1971)
[3] Baranovskii E. P., Mathematica 2 pp 18– (1999)
[4] Baranovskii E. P., Mathematical Notes 68 (6) pp 704– (2000) · Zbl 1008.52014
[5] Christof T., Internat. J. Comput. Geom. Appl. 11 (4) pp 423– (2001) · Zbl 1074.68635
[6] Conway J. H., Sphere Packings, Lattices and Groups, (1999)
[7] De Simone C., Discrete Mathematics 127 pp 105– (1994) · Zbl 0799.90099
[8] Deza A., Polytopes: Abstract, Convex and Computational pp 359– (1994)
[9] Deza M., The Quarterly Journal of Mathematics Oxford 44 (2) pp 399– (1993) · Zbl 0795.05120
[10] Deza M., Geometry of Cuts and Metrics. (1997) · Zbl 0885.52001
[11] Deza M., Sets, Graphs and Numbers, Budapest (Hungary), 1991 pp 157– (1992)
[12] Deza M., Combinatorica 13 pp 397– (1993) · Zbl 0801.52009
[13] Dutour, M. ”The Six Dimensional Delaunay Polytopes.”. Proc. Int. Conference on Arithmetics and Combinatorics. [Dutour 02], To appear in · Zbl 1046.52009
[14] Dutour M., ”The Gosset Polytope and the Hypermetric Cone on Eight Vertices.” · Zbl 1390.68716
[15] Dutour M., ”The Extreme Rays of the Hypermetric Cone HYP7.” (2003)
[16] Fedorov E. S., Elements of the Theory of Figures (in Russian) (1885)
[17] Fukuda K., ”The cdd Program.” (2003)
[18] Grishukhin V. P., European Journal of Combinatorics 11 pp 115– (1990)
[19] Grishukhin V. P., European Journal of Combinatorics 13 pp 153– (1992) · Zbl 0760.05058
[20] Kononenko P. G., Theses (kandidatskaia dissertacia), in: ”Affine Types of L-Polytopes of Five-Dimensional Lattices” (in Russian) (1999)
[21] Kononenko P. G., Mat. Zametki 71 (3) pp 412– (2002)
[22] Ryshkov S. S., Voronoi’s Impact on Modern Science pp 115– (1998)
[23] Ryshkov S. S., Canad. J. Math. 39 (4) pp 794– (1987) · Zbl 0643.10025
[24] Ryshkov S. S., Canad. J. Math. 40 (5) pp 1058– (1988) · Zbl 0653.10027
[25] McKay B., ”The nauty Program.” (2003)
[26] Voronoi G. F., J. fur die reine und angewandte Mathematik 134 pp 198– (1908)
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