Problems from CGCS

Most of these problems were presented at the conference on Combinatorics, Geometry and Computer Science, held at CIRM, Luminy, 2-4 May 2007. I have added one problem on a similar theme after the meeting. The problems have been arranged so that those with similar themes occur together as far as possible. 1. Nearly round polytopes Communicated by Komei Fukuda. We call a polytope centered if it contains the origin in its relative interior. Nesterov [17] has recently proved that the Minkowski sum of a centered full-dimensional polytope (and more generally a centered full-dimensional compact convex body) with a properly scaled copy of its dual gives a set whose asphericity is at most the square root of that of the initial polytope. The asphericity of a set is here defined as the ratio of the diameter of its smallest enclosing ball to that of its largest enclosed ball. Thus, summing a polytope with its own dual has a strong rounding effect. For this reason, the Minkowski sum P + αP , will be called a Nesterov rounding of a polytope