On Combinatorics of Voronoi Polytopes for Perturbations of the Dual Root Lattices

Abstract

The Voronoi conjecture on parallelohedra claims that for every convex polytope P that tiles Euclidean d-dimensional space with translations there exists a d-dimensional lattice such that P and the Voronoi polytope of this lattice are affinely equivalent. The Voronoi conjecture is still open for the general case but it is known that some combinatorial restrictions for the face structure of P ensure that the Voronoi conjecture holds for P. In this article, we prove that if P is the Voronoi polytope of one of the dual root lattices $D_d^{*},\hspace{0.17em}{E}_6^{*},\hspace{0.17em}{E}_7^{*}$ or $E_8^{*}=E_8$ or their small perturbations, then every parallelohedron combinatorially equivalent to P in strong sense satisfies the Voronoi conjecture.

KEYWORDS:

• Parallelohedra
• Root lattices
• Voronoi conjecture
• Venkov graph

Acknowledgments

This work was finished while the author was a visiting professor at IST Austria. The author is thankful to IST Austria and the group of Herbert Edelsbrunner for hospitality and support.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Author’s research is partially supported by the Alexander von Humboldt Foundation.