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This article is cited in 4 scientific papers (total in 4 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.
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
Linking options:
https://www.mathnet.ru/eng/sm3698https://doi.org/10.1070/SM2006v197n10ABEH003805 https://www.mathnet.ru/eng/sm/v197/i10/p15
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Abstract page: | 497 | Russian version PDF: | 225 | English version PDF: | 13 | References: | 54 | First page: | 1 |
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