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

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

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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

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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• http://mi.mathnet.ru/eng/msb3698
• https://doi.org/10.4213/sm3698
• http://mi.mathnet.ru/eng/msb/v197/i10/p15

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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
2. A. N. Magazinov, “Voronoi's conjecture for extensions of Voronoi parallelohedra”, Russian Math. Surveys, 69:4 (2014), 763–764
3. A. A. Gavrilyuk, “Geometry of lifts of tilings of Euclidean spaces”, Proc. Steklov Inst. Math., 288 (2015), 39–55
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