Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices
نویسندگان
چکیده
A polytope D, whose vertices belong to a lattice of rank d, is Delaunay if it can be circumscribed by an ellipsoid E with interior free of lattice points, and so that the vertices of D are the only lattice points on the quadratic surface E. If in addition E is uniquely determined by D, we call D a perfect Delaunay polytope. Thus, in the perfect case, the lattice points on E, which are the vertices of D, uniquely determine the quadratic surface E. We have been able to construct infinite sequences of perfect Delaunay polytopes, one perfect polytope in each successive dimension starting at some initial dimension; we have been able to construct an infinite number of such sequences. Perfect Delaunay polytopes play an important role in the theory of Delaunay polytopes and in Voronoi’s theory of lattice types.
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