Semigroups of polygons whose vertices define a~centered partition of~$\mathbb R^n$

V. M. Gichev ; I. A. Zubareva ; E. A. Mescheryakov

Matematičeskie trudy, Tome 15 (2012) no. 1, pp. 55-73

Résumé

A partition $\mathfrak F$ of a Euclidean space into finite subsets has subgroup property $\mathsf{SP}$ if the family of the convex hulls of the leaves of $\mathfrak F$ constitutes a subgroup with respect to the Minkowski addition. If $\mathfrak F$ consists of orbits of a finite linear groups then $\mathsf{SP}$ is equivalent to the fact that the group is a Coxeter group. In this article, this assertion is proved only under the assumption of continuity and centrality of $\mathfrak F$ (this means that every leaf is inscribed in some sphere centered at zero). An example is given of a noncentered partition satisfying $\mathsf{SP}$ (such partitions cannot be Coxeter partitions).