Voronoi conjecture for five-dimensional parallelohedra

TL;DR

This work proves the Voronoi conjecture for five-dimensional parallelohedra by combining a rigorous local-to-global combinatorial analysis of dual cells with a freeness/canonical-scaling framework. The authors show that any 5D parallelohedron either possesses a free direction, which guarantees the conjecture, or admits a canonical scaling ensuring the tiling is affinely equivalent to a Dirichlet-Voronoi polytope of a 5D lattice. A detailed case analysis centered on dual 3-cells (tetrahedra, octahedra, pyramids, prisms) and their associated 4-cells demonstrates that all possible local configurations lead to the conjecture, often via freeness outcomes or by establishing canonical scaling through a gain function. The result confirms the completeness of the 5D Dirichlet-Voronoi parallelohedra list and clarifies the limitations of prior approaches (e.g., Engel’s zone-contraction program) for higher dimensions, while outlining key open questions for dimensions six and higher.

Abstract

We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely, we show that if a convex five-dimensional polytope $P$ tiles $\mathbb R^5$ 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 $P$.

Figures (5)

• Figure 1: Five three-dimensional parallelohedra: hexagonal prism, rhombic dodecahedron, parallelepiped, elongated dodecahedron, and truncated octahedron.
• Figure 2: Two types of dual 2-cells. Original parallelohedra are black polygons and dual cells are red.
• Figure 3: An illustration for the proof of Lemma \ref{['lem:pr-pr-py']}. The face $G$ with $\mathcal{D}(G)=XYY'X$ and its triangular faces $xyz$ and $xyt$ with prismatic dual cells. We put dual cells of two-dimensional faces inside corresponding triangles and show only additional points corresponding to edges. Dual cells are shown in red.
• Figure 4: An illustration for the proof of Lemma \ref{['lem:pr-py-py']}. The face $G$ with $\mathcal{D}(G)=XYY'X$ and its triangular faces $xyz$ and $xyt$ with prismatic dual cells. We put dual cells of two-dimensional faces inside corresponding triangles and show only additional points corresponding to edges. Dual cells are shown in red.
• Figure 5: An illustration for the proof of Lemma \ref{['lem:coherent']}. The face $G$ with $\mathcal{D}(G)=XYY'X$ and its faces $G_1$ and $G_2$ with a common vertex $V$. Two paths from $G_1$ to $G_2$ around $V$ pass through triangular faces $H_1$ and $H_2$ with prismatic dual cells.

Theorems & Definitions (75)

• Conjecture : G. Voronoi
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5: G. Voronoi
• Theorem 2.6: O. Zhitmorski
• Theorem 2.7
• Theorem 2.8: A. Ordine
• Definition 2.9