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A note on the Hermite–Rankin constant

Kazuomi Sawatani, Takao Watanabe, Kenji Okuda (2010)

Journal de Théorie des Nombres de Bordeaux

We generalize Poor and Yuen’s inequality to the Hermite–Rankin constant γ n , k and the Bergé–Martinet constant γ n , k . Moreover, we determine explicit values of some low- dimensional Hermite–Rankin and Bergé–Martinet constants by applying Rankin’s inequality and some inequalities proven by Bergé and Martinet to explicit values of γ 5 , γ 7 , γ 4 , 2 and γ n ( n 8 ).

Delaunay polytopes derived from the Leech lattice

Mathieu Dutour Sikirić, Konstantin Rybnikov (2014)

Journal de Théorie des Nombres de Bordeaux

A Delaunay polytope in a lattice L is perfect if any affine transformation that preserve its Delaunay property is a composite of an homothety and an isometry. Perfect Delaunay polytopes are rare in low dimension and here we consider the ones that one can get in lattice that are sections of the Leech lattice.By doing so we are able to find lattices with several orbits of perfect Delaunay polytopes. Also we exhibit Delaunay polytopes which remain Delaunay in some superlattices. We found perfect Delaunay...

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