The second Voronoi conjecture on parallelohedra for zonotopes

TL;DR

This paper proves the second Voronoi conjecture for zonotope parallelohedra, demonstrating that in a face-to-face tiling, certain vectors form a basis of the tiling's lattice.

Contribution

It establishes the second Voronoi conjecture specifically for zonotope parallelohedra, a case not previously proven.

Findings

Vectors connecting centers of zonotopes with common facets form a basis of the lattice.

The proof applies to face-to-face tilings of Euclidean space with zonotopes.

Supports the conjecture's validity in the context of zonotope tilings.

Abstract

We prove the second Voronoi conjecture on parallelohedra for zonotope. We show that for a given face-to-face tiling of d-dimensional Euclidean space into parallel copies of zonotope Z there are d vectors, connecting centers of zonotopes with common facet, that are basis of the correspondent lattice of the tiling.