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.
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