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).


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


Zbl 0885.52001


cdd; nauty
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