We study integer programming instances over polytopes P (A, b) = {x : Ax ≤ b} where the constraint matrices A are random – the rows of the constraint matrices are chosen i.i.d. from a spherically symmetric distribution. We address the radius of the largest inscribed ball that guarantees integer feasibility of such random polytopes with high probability. We show that for m = 2 √ , there exist co...

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 co...

The thickness (or minimum width) of a convex body K in Euclidean space E, d ≥ 2, is the minimal distance between two distinct parallel supporting hyperplanes of K. The body K is said to be reduced if no proper convex subset of K has the same thickness. It is natural to study the notions of thickness and reduced body also in Minkowski spaces (that is, in real normed linear spaces of finite dimen...

we consider the semigroup $s$ of highest weights appearing in tensor powers $v^{otimes k}$ of a finite dimensional representation $v$ of a connected reductive group. we describe the cone generated by $s$ as the cone over the weight polytope of $v$ intersected with the positive weyl chamber. from this we get a description for the asymptotic of the number of highest weights appearing in $v^{otime...

We study metric properties of convex bodies B and their polars B, where B is the convex hull of an orbit under the action of a compact group G. Examples include the Traveling Salesman Polytope in polyhedral combinatorics (G = Sn, the symmetric group), the set of nonnegative polynomials in real algebraic geometry (G = SO(n), the special orthogonal group), and the convex hull of the Grassmannian ...