Monday, November 11, 2013
Freie Universität Berlin
Institut für Informatik
14195 Berlin Berlin
Lecture - 14:15
A stereohedron is a polytope that tiles Euclidean space by the action of a discrete group of isometries, necessarily a crystallographic group. Delone (1961) showed that a stereohedron of dimension d for a group with a translational cosets (so-called ``aspects'') cannot have more than 2d(a+1)-2 facets. In dimension three Delone's bound allows up to 390 facets, for groups having as aspects all the 48 symmetries of a regular cube. In contrast, the stereohedron with the maximum number of facets known (found by Engel, 1981) has only 38 facets.
In this talk I will report on joint work with D. Bochis and P. Sabariego in which we show that Dirichlet stereohedra cannot have more than 92 facets. Here, Dirichlet (or Voronoi) stereohedra are the particular stereohedra obtained as Voronoi regions of an orbit of points of the crystallographic group. We will focus on cubic groups, which are the most complicated ones and where our bound is worst. Here use the recent classification of cubic groups by Conway et al (2001). Our method combines general principles with case-by-case study, and some computer calculations.
Colloquium - 16:00
We present a complete 3-dimensional Blaschke-Santaló diagram for planar convex bodies with respect to the four classical magnitudes inner and outer radius, diameter and (minimal) width in Euclidean spaces. This is joint work with René Brandenberg.