Properties of parallelotopes equivalent to Voronoi’s conjecture

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, that is to say a Voronoi polytope. We give several properties of a parallelotope and prove that each of them is equivalent to it being an affine image of a Voronoi polytope.