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Perfect Delaunay Polytopes in Low Dimensions

Mathieu Dutour·Robert Erdahl·Konstantin Rybnikov
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Abstract

A lattice Delaunay polytope is known as perfect if the only ellipsoid, that can be circumscribed about it, is its Delaunay sphere. Perfect Delaunay polytopes are in one-to-one correspondence with arithmetic equivalence classes of positive quadratic functions on the n-dimensional integral lattice that can be recovered, up to a scale factor, from the representations of its minimum. We develop a structural theory of such polytopes and describe all known perfect Delaunay polytopes in dimensions one through eight. We suspect that this list is complete.

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Mathieu Dutour, Robert Erdahl, Konstantin Rybnikov. 2007-02-06. Perfect Delaunay Polytopes in Low Dimensions. https://arxiv.org/abs/math/0702136

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