Minkowski sum of a parallelotope and a segment

V. P. Grishukhin

Sbornik. Mathematics, Tome 197 (2006) no. 10, pp. 1417-1433

Résumé

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.