V. M. Gichev, I. A. Zubareva, E. A. Mescheryakov, “Semigroups of polygons whose vertices define a centered partition of <nobr>$\mathbb R^n$</nobr>”, Mat. Tr., 2012, Volume 15, Number 1,Pages <nobr>55

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

V. M. Gichev^a, I. A. Zubareva^a, E. A. Mescheryakov^b

^a Sobolev Institute of Mathematics, Omsk Division, Omsk, Russia
^b Omsk State University, Omsk, Russia

Abstract: 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).

Key words: semigroups of polygon, Coxeter groups.

UDC: 514.1

Received: 26.04.2011