Cutting Convex Polytopes by Hyperplanes

Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the original polytope are hereditary to its subpolytopes obtained by a cut. In this work, we devote our attention to all the separating hyperplanes for some given polytope (integral and convex) and study the existence and classification of such hyperplanes. We prove the existence of separating hyperplanes for the order and chain polytopes for any finite posets that are not a single chain, and prove there are no such hyperplanes for any Birkhoff polytopes. Moreover, we give a complete separating hyperplane classification for the unit cube and its subpolytopes obtained by one cut, together with some partial classification results for order and chain polytopes.

Faces of Birkhoff Polytopes

The Birkhoff polytope $B_n$ is the convex hull of all $(n\times n)$ permutation matrices, i.e., matrices where precisely one entry in each row and column is one, and zeros at all other places. This is a widely studied polytope with various applications throughout mathematics.In this paper we study combinatorial types $\mathcal L$ of faces of a Birkhoff polytope. The Birkhoff dimension $\mathrm{bd}(\mathcal L)$ of $\mathcal L$ is the smallest $n$ such that $B_n$ has a face with combinatorial type $\mathcal L$.By a result of Billera and Sarangarajan, a combinatorial type $\mathcal L$ of a $d$-dimensional face appears in some $\mathcal B_k$ for $k\le 2d$, so $\mathrm{bd}(\mathcal L)\le 2d$. We will characterize those types with $\mathrm{bd}(\mathcal L)\ge 2d-3$, and we prove that any type with $\mathrm{bd}(\mathcal L)\ge d$ is either a product or a wedge over some lower dimensional face. Further, we computationally classify all $d$-dimensional combinatorial types for $2\le d\le 8$.

Perturbation of transportation polytopes

International audience We describe a perturbation method that can be used to compute the multivariate generating function (MGF) of a non-simple polyhedron, and then construct a perturbation that works for any transportation polytope. Applying this perturbation to the family of central transportation polytopes of order $kn \times n$, we obtain formulas for the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order $kn \times n$. Nous décrivons une méthode de perturbation qui peut être utilisée pour calculer la fonction génératrice multivariée (MGF) d'un polyèdre non-simple, et ensuite construire une perturbation qui fonctionne pour tout polytope de transport. Appliquant cette perturbation à la famille des centraux de transport polytopes de l'ordre $kn \times n$, nous obtenons des formules pour le MGF du polytope. Les formules que nous obtenons sont énumérées par les objets combinatoires. Un cas spécial des formules récupère les résultats sur des polytopes de Birkhoff donnés par l'auteur et De Loera et Yoshida. Nous récupérons également la formule pour le nombre de sommets maximum des de transport polytopes d'ordre $kn \times n$.