Covering the Permutohedron by Affine Hyperplanes
The vertices of the permutohedron can be covered by one hyperplane in the $n$ dimensional affine space. We prove here that any set of hyperplanes that covers all the vertices of the permutohedron but one contains at least ${n\choose 2}$ hyperplanes. Our proof is based on a new variant of the Combinatorial Nullstellensatz.