A Class of Affinely Equivalent Voronoi Parallelohedra

A. A. Gavrilyuk

Matematičeskie zametki, Tome 95 (2014) no. 5, pp. 697-707.

Résumé

Given any parallelohedron $P$, its affine class $\mathscr A(P)$, i.e., the set of all parallelohedra affinely equivalent to it, is considered. Does this affine class contain at least one Voronoi parallelohedron, i.e., a parallelohedron which is a Dirichlet domain for some lattice? This question, more commonly known as Voronoi's conjecture, has remained unanswered for more than a hundred years. It is shown that, in the case where the subset of Voronoi parallelohedra in $\mathscr A(P)$ is nonempty, this subset is an orbifold, and its dimension (as a real manifold with singularities) is completely determined by its combinatorial type; namely, it is equal to the number of connected components of the so-called Venkov subgraph of the given parallelohedron. Nevertheless, the structure of this orbifold depends not only on the combinatorial properties of the parallelohedron but also on its affine properties.

Détail

Mots-clés : parallelohedron, Voronoi parallelohedron
Keywords: affinely equivalent parallelohedra, Venkov graph, Venkov subgraph, orbifold of Voronoi parallelohedra.

Bibliographie

[1] E. S. Fedorov, Nachala ucheniya o figurakh, Sankt-Peterburg, 1885

[2] H. Minkowski, “Allgemeine Lehersätze über konvexe Polyeder”, Göott. Nachr., 1897 (1897), 198–219 | Zbl

[3] B. A. Venkov, “Ob odnom klasse evklidovykh mnogogrannikov”, Vestn. Leningradsk. un-ta. Ser. matem., fiz., khim., 1954, no. 9, 11–31 | MR

[4] P. McMullen, “Convex bodies which tile space by translation”, Mathematika, 27:1 (1980), 113–121 | DOI | MR | Zbl

[5] N. P. Dolbilin, “The extension theorem”, Discrete Math., 221:1-3 (2000), 43–59 | DOI | MR | Zbl

[6] N. P. Dolbilin, V. S. Makarov, “Teorema o prodolzhenii v teorii pravilnykh razbienii i ee prilozheniya”, Diskretnaya geometriya i geometriya chisel, Tr. MIAN, 239, Nauka, M., 2002, 146–169 | MR | Zbl

[7] G. Voronoi, “Nouvelles applications des paramétres continus á la theorie des formes quadratiques. Deuxième mémoire. Recherches sur les paralléloédres primitifs”, J. für Math., 136 (1909), 67–178 | Zbl

[8] O. Zitomirskij, “Vershärfung eines Satzes von Woronoi”, Zhurn. Leningradsk. fiz.-matem. ob-va, 2:2 (1929), 131–151

[9] A. Ordine, Proof of the Voronoi conjecture on parallelotopes in a new special case, PhD Thesis, Queen's University, Kingston, 2005 | MR

[10] A. Garber, A. Gavrilyuk, A. Magazinov, “The Voronoi conjecture for parallelohedra with simply connected $\delta$-surface”, Discrete Comput. Geom. (to appear) , arXiv: math.MG/1212.1019v1

[11] R. M. Erdahl, “Zonotopes, dicings, and Voronoi's conjecture on parallelohedra”, European J. Combin., 20:6 (1999), 527–549 | DOI | MR | Zbl

[12] L. Michel, S. S. Ryshkov, M. Senechal, “An extension of Voronoï's theorem on primitive parallelotopes”, European J. Combin., 16 (1995), 59–63 | DOI | MR | Zbl

[13] N. Dolbilin, J.-I. Itoh, C. Nara, “Affine equivalent classes of parallelohedra”, Computational Geometry, Graphs and Applications, Lecture Notes in Comput. Sci., 7033, Springer-Verlag, Heidelberg, 2011, 55–60 | DOI | MR | Zbl

[14] N. P. Dolbilin, “Paralleloedry: retrospektiva i novye rezultaty”, Tr. MMO, 73, no. 2, MTsNMO, M., 2012, 259–276 | Zbl

[15] N. P. Dolbilin, “Svoistva granei paralleloedrov”, Geometriya, topologiya i matematicheskaya fizika. II, Tr. MIAN, 266, MAIK, M., 2009, 112–126 | MR | Zbl

[16] G. C. Shephard, “Polytopes with centrally symmetric faces”, Canad. J. Math., 19 (1967), 1206–1213 | MR | Zbl

[17] G. M. Ziegler, Lectures on Polytopes, Grad. Texts in Math., 152, Springer-Verlag, New-York, 1995 | MR | Zbl

[18] W. P. Thurston, The Geometry and Topology of Three-Manifolds, Princeton Univ. Lectures Notes, Princeton Univ., 1980