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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1906.05193 |
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| _version_ | 1866929704044331008 |
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| author | Garber, Alexey |
| author_facet | Garber, Alexey |
| contents | 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 $P$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_05193 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Voronoi conjecture for five-dimensional parallelohedra Garber, Alexey Combinatorics 52B20, 52C07 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 $P$. |
| title | Voronoi conjecture for five-dimensional parallelohedra |
| topic | Combinatorics 52B20, 52C07 |
| url | https://arxiv.org/abs/1906.05193 |