On Voronoi’s conjecture for four- and five-dimensional parallelohedra. (English. Russian original) Zbl 1489.52012
Russ. Math. Surv. 77, No. 1, 174-176 (2022); translation from Usp. Mat. Nauk 77, No. 1, 185-186 (2022).
Consider some tiling of the Euclidean space \(\mathbb{R}^d\) such that all the \(d\)-dimensional cells in this tiling are translates of a given polytope \(P\). G. Voronoï’s conjecture [J. Reine Angew. Math. 133, 97–178 (1908; JFM 38.0261.01); ibid. 134, 198–287 (1908; JFM 39.0274.01)] asks whether such a tiling is always, up to some affine transformation \(\mathbb{R}^d\rightarrow\mathbb{R}^d\), the Voronoi diagram of a lattice.
A proof of this conjecture was given when \(d=4\) by B. Delaunay [Bull. Acad. Sci. URSS 2, 79–110 (1929; JFM 56.1120.02)]. Since Delaunay’s paper, significant effort lead to proving the conjecture in special cases as for instance when \(P\) is a zonotope [R. M. Erdahl, Eur. J. Comb. 20, No. 6, 527–549 (1999; Zbl 0938.52016)]. However, all higher dimensional cases remained open in general, until a proof when \(d=5\) was finally announced in 2019 by Alexey Garber and Alexander Magazinov.
In this short article, the authors explain how the argument of their \(5\)-dimensional proof can be adapted into a \(4\)-dimensional proof alternative to Delaunay’s. They also provide an outline of the \(5\)-dimensional argument.
A proof of this conjecture was given when \(d=4\) by B. Delaunay [Bull. Acad. Sci. URSS 2, 79–110 (1929; JFM 56.1120.02)]. Since Delaunay’s paper, significant effort lead to proving the conjecture in special cases as for instance when \(P\) is a zonotope [R. M. Erdahl, Eur. J. Comb. 20, No. 6, 527–549 (1999; Zbl 0938.52016)]. However, all higher dimensional cases remained open in general, until a proof when \(d=5\) was finally announced in 2019 by Alexey Garber and Alexander Magazinov.
In this short article, the authors explain how the argument of their \(5\)-dimensional proof can be adapted into a \(4\)-dimensional proof alternative to Delaunay’s. They also provide an outline of the \(5\)-dimensional argument.
Reviewer: Lionel Pournin (Paris)
MSC:
| 52B11 | \(n\)-dimensional polytopes |
| 52C07 | Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry) |
| 52C22 | Tilings in \(n\) dimensions (aspects of discrete geometry) |
Online Encyclopedia of Integer Sequences:
Number of combinatorial types of n-dimensional parallelohedra.References:
| [1] | Voronoi, G.; Voronoi, G.; Voronoi, G., J. Reine Angew. Math.. J. Reine Angew. Math.. J. Reine Angew. Math., 1909, 136, 67-178 (1909) · JFM 40.0267.17 · doi:10.1515/crll.1909.136.67 |
| [2] | Delaunay (Delone), B.; Delaunay (Delone), B., Izv. Akad. Nauk SSSR Ser. VII. Otd. Fiz.-Mat. Nauk. Izv. Akad. Nauk SSSR Ser. VII. Otd. Fiz.-Mat. Nauk, 2, 147-164 (1929) |
| [3] | Garber, A.; Magazinov, A., Voronoi conjecture for five-dimensional parallelohedra (2020) |
| [4] | Goodman, J. E.; O’Rourke, J.; ; C. D. T\'{o}th (eds.), Handbook of discrete and computational geometry, xxi+1927 pp. (2017), CRC Press: CRC Press, Boca Raton, FL · Zbl 1375.52001 · doi:10.1201/9781315119601 |
| [5] | Dolbilin, N. P.; Dolbilin, N. P., Trans. Moscow Math. Soc., 73, 2, 207-220 (2012), Moscow Center for Continuous Mathematical Education: Moscow Center for Continuous Mathematical Education, Moscow · Zbl 1271.52013 · doi:10.1090/S0077-1554-2013-00208-3 |
| [6] | Minkowski, H., Nachr. Ges. Wiss. Göttingen. Math.-Phys. Kl., 1897, 198-219 (1897) · JFM 28.0427.01 |
| [7] | Venkov, B. A., Vestn. Leningrad. Univ. Ser. Mat. Fiz. Khim., 9, 2, 11-31 (1954) |
| [8] | McMullen, P., Mathematika, 27, 1, 113-121 (1980) · Zbl 0432.52016 · doi:10.1112/S0025579300010007 |
| [9] | Zitomirskij (Zhitomirskii), O. K., Zh. Leningrad. Fiz.-Mat. Obshch., 2, 2, 131-151 (1929) |
| [10] | Dutour Sikiri\'c, M.; Grishukhin, V.; ; Magazinov, A., European J. Combin., 42, 49-73 (2014) · Zbl 1314.52010 · doi:10.1016/j.ejc.2014.05.005 |
| [11] | Garber, A., Ann. Comb., 21, 4, 551-572 (2017) · Zbl 1431.52016 · doi:10.1007/s00026-017-0366-9 |
| [12] | Ordine, A., Proof of the Voronoi conjecture on parallelotopes in a new special case, 131 pp. (2005), Queen’s Univ.: Queen’s Univ., Canada |
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