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Mat. Sb., 2004, Volume 195, Number 5, Pages 59–78 (Mi msb821)  

This article is cited in 10 scientific papers (total in 10 papers)

Parallelotopes of non-zero width

V. P. Grishukhin

Central Economics and Mathematics Institute, RAS

Abstract: In 1959, Venkov introduced a concept of polytope of non-zero width in the direction of a subspace and studied parallelotopes of non-zero width. In the present paper properties of a parallelotope of non-zero width in the direction of a straight line are investigated. In particular, it is proved that a parallelotope of non-zero width in the direction of a straight line is the Minkowski sum of a parallelotope of width zero and a segment of this line. Necessary and sufficient conditions ensuring that the sum of a parallelotope and a line segment is again a parallelotope are presented.

DOI: https://doi.org/10.4213/sm821

Full text: PDF file (321 kB)
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English version:
Sbornik: Mathematics, 2004, 195:5, 669–686

Bibliographic databases:

UDC: 511.9
MSC: Primary 52B11; Secondary 52C22
Received: 20.03.2003

Citation: V. P. Grishukhin, “Parallelotopes of non-zero width”, Mat. Sb., 195:5 (2004), 59–78; Sb. Math., 195:5 (2004), 669–686

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. P. Grishukhin, “Free and Nonfree Voronoi Polyhedra”, Math. Notes, 80:3 (2006), 355–365  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. V. P. Grishukhin, “Minkowski sum of a parallelotope and a segment”, Sb. Math., 197:10 (2006), 1417–1433  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. Horváth Á.G., “On the connection between the projection and the extension of a parallelotope”, Monatsh. Math., 150:3 (2007), 211–216  crossref  mathscinet  zmath  isi
    4. Deza M., Grishukhin V.P., “More about the 52 four-dimensional parallelotopes”, Taiwanese J. Math., 12:4 (2008), 901–916  crossref  mathscinet  zmath  isi  elib
    5. Dutour Sikirić M., Grishukhin V., “The decomposition of the hypermetric cone into $L$-domains”, European J. Combin., 30:4 (2009), 853–865  crossref  mathscinet  zmath  isi  elib
    6. A. Végh, “On the orthogonal projections of Dirichlet–Voronoi cells of lattices”, Beitr. Algebra Geom., 52:2 (2011), 487–493  crossref  mathscinet  zmath
    7. V. P. Grishukhin, “Delaunay and Voronoi polytopes of the root lattice $E_7$ and of the dual lattice $E_7^*$”, Proc. Steklov Inst. Math., 275 (2011), 60–77  mathnet  crossref  mathscinet  isi  elib  elib
    8. V. P. Grishukhin, “The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice $E_7$”, Sb. Math., 203:11 (2012), 1571–1588  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. M. D. Sikirić, V. Grishukhin, A. Magazinov, “On the sum of a parallelotope and a zonotope”, European J. Combin., 42 (2014), 49–73  crossref  mathscinet  zmath  isi
    10. V. P. Grishukhin, V. I. Danilov, “Lifting of parallelohedra”, Sb. Math., 210:10 (2019), 1434–1455  mathnet  crossref  crossref  adsnasa  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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