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Properties of parallelotopes equivalent to Voronoi's conjecture

Michel Deza·Viacheslav Grishukhin
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Abstract

A parallelotope is a polytope whose translation copies fill space without gaps and intersections by interior points. Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, which is a Voronoi polytope. We give several properties of a parallelotope and prove that each of them is equivalent to it is an affine image of a Voronoi polytope.

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Michel Deza, Viacheslav Grishukhin. 2003-07-11. Properties of parallelotopes equivalent to Voronoi's conjecture. https://arxiv.org/abs/math/0307170

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