Voronoi conjecture for five-dimensional parallelohedra

Authors

Alexey Garber, The University of Texas Rio Grande ValleyFollow
Alexander Magazinov

Document Type

Article

Publication Date

2-10-2025

Abstract

We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely, we show that if a convex five-dimensional polytope P tiles ℝ5 with translations, then P is an affine image of the Dirichlet-Voronoi polytope for a five-dimensional lattice.
Our proof is based on an exhaustive combinatorial analysis of possible dual 3-cells and incident dual 4-cells encoding local structures around two-dimensional faces of five-dimensional parallelohedron P and their edges aiming to prove existence of a free direction for P paired with new properties established for parallelohedra (in any dimension) that have a free direction that guarantee the Voronoi conjecture for P.

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Recommended Citation

Garber, Alexey. "Voronoi conjecture for five-dimensional parallelohedra." Inventiones mathematicae (2025): 1-45. https://doi.org/10.1007/s00222-025-01325-0

Publication Title

Inventiones mathematicae

DOI

10.1007/s00222-025-01325-0