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 Mat. Zametki, 2006, Volume 80, Issue 3, Pages 367–378 (Mi mz2822)

Free and Nonfree Voronoi Polyhedra

V. P. Grishukhin

Central Economics and Mathematics Institute, RAS

Abstract: The Voronoi polyhedron of some point $v$ of a translation lattice is the closure of the set of points in space that are closer to $v$ than to any other lattice points. Voronoi polyhedra are a special case of parallelohedra, i.e., polyhedra whose parallel translates can fill the entire space without gaps and common interior points. The Minkowski sum of a parallelohedron with a segment is not always a parallelohedron. A parallelohedron $P$ is said to be free along a vector $e$ if the sum of $P$ with a segment of the line spanned by $e$ is a parallelohedron. We prove a theorem stating that if the Voronoi polyhedron $P_V(f)$ of a quadratic form $f$ is free along some vector, then the Voronoi polyhedron $P_V(g)$ of each form $g$ lying in the closure of the L-domain of $f$ is also free along some vector. For the dual root lattice $E_6^*$ and the infinite series of lattices $D_{2m}^+$, $m\geqslant 4$, we prove that their Voronoi polyhedra are nonfree in all directions.

Keywords: parallelohedron, Voronoi polyhedron, Delaunay polyhedron, Minkowski sum, quadratic form, L-domain, Gram matrix.

DOI: https://doi.org/10.4213/mzm2822

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English version:
Mathematical Notes, 2006, 80:3, 355–365

Bibliographic databases:

UDC: 511.9
Revised: 10.01.2006

Citation: V. P. Grishukhin, “Free and Nonfree Voronoi Polyhedra”, Mat. Zametki, 80:3 (2006), 367–378; Math. Notes, 80:3 (2006), 355–365

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz2822
• https://doi.org/10.4213/mzm2822
• http://mi.mathnet.ru/eng/mz/v80/i3/p367

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Dutour Sikirić M., Grishukhin V., “The decomposition of the hypermetric cone into $L$-domains”, European J. Combin., 30:4 (2009), 853–865
2. V. G. Zhuravlev, “Exchanged toric developments and bounded remainder sets”, J. Math. Sci. (N. Y.), 184:6 (2012), 716–745
3. V. G. Zhuravlev, “Teorema Gekke: forma i ideya”, Chebyshevskii sb., 12:1 (2011), 79–92
4. Sikiric M.D., Grishukhin V., Magazinov A., “On the Sum of a Parallelotope and a Zonotope”, Eur. J. Comb., 42 (2014), 49–73
5. V. P. Grishukhin, V. I. Danilov, “Lifting of parallelohedra”, Sb. Math., 210:10 (2019), 1434–1455
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