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Mat. Sb., 2006, Volume 197, Number 10, Pages 15–32 (Mi msb3698)  

This article is cited in 3 scientific papers (total in 3 papers)

Minkowski sum of a parallelotope and a segment

V. P. Grishukhin

Central Economics and Mathematics Institute, RAS

Abstract: Not every parallelotope $P$ is such that the Minkowski sum $P+S_e$ of $P$ with a segment $S_e$ of the straight line along a vector $e$ is a parallelotope. If $P+S_e$ is a parallelotope, then $P$ is said to be free along $e$. The parallelotope $P+S_e$ is not always a Voronoĭ polytope. The well-known Voronoĭ conjecture states that every parallelotope is affinely equivalent to a Voronoĭ polytope. An attempt is made to prove Voronoĭ's conjecture for $P+S_e$. For that a class $\mathscr P(e)$ of canonically defined parallelotopes that are free along $e$ is introduced. It is proved that $P+S_e$ is affinely equivalent to a Voronoĭ polytope if and only if $P$ is a direct sum of parallelotopes of class $\mathscr P(e)$.
This simple case of the proof of Voronoĭ's conjecture is an instructive example for understanding the general case.
Bibliography: 10 titles.

DOI: https://doi.org/10.4213/sm3698

Full text: PDF file (500 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2006, 197:10, 1417–1433

Bibliographic databases:

UDC: 511.6+514.174.6
MSC: Primary 52C22; Secondary 51M20, 52B11, 52B20, 52C07
Received: 19.05.2005 and 23.03.2006

Citation: V. P. Grishukhin, “Minkowski sum of a parallelotope and a segment”, Mat. Sb., 197:10 (2006), 15–32; Sb. Math., 197:10 (2006), 1417–1433

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. Mathieu Dutour Sikirić, Viacheslav Grishukhin, Alexander Magazinov, “On the sum of a parallelotope and a zonotope”, European Journal of Combinatorics, 42 (2014), 49  crossref  mathscinet  zmath
    2. A. N. Magazinov, “Voronoi's conjecture for extensions of Voronoi parallelohedra”, Russian Math. Surveys, 69:4 (2014), 763–764  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. A. A. Gavrilyuk, “Geometry of lifts of tilings of Euclidean spaces”, Proc. Steklov Inst. Math., 288 (2015), 39–55  mathnet  crossref  crossref  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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