On a Special Class of Hyper-Permutahedra

Minkowski sums of simplices in ${\mathbb{R}}^n$ form an interesting class of polytopes that seem to emerge in various situations. In this paper we discuss the Minkowski sum of the simplices $\Delta_{k-1}$ in ${\mathbb{R}}^n$ where $k$ and $n$ are fixed, their flags and some of their face lattice structure. In particular, we derive a closed formula for their exponential generating flag function. These polytopes are simple, include both the simplex $\Delta_{n-1}$ and the permutahedron $\Pi_{n-1}$, and form a Minkowski basis for more general permutahedra.

Affine and toric arrangements

International audience We extend the Billera―Ehrenborg―Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky's fundamental results on the number of regions. Nous étendons l'opérateur de Billera―Ehrenborg―Readdy entre le trellis d'intersection et la treillis de faces d'un arrangement hyperplans centraux aux arrangements affines et toriques. Pour les arrangements toriques, nous généralisons aussi les résultats fondamentaux de Zaslavsky sur le nombre de régions.

Lawrence Polytopes

In 1980 Jim Lawrence suggested a construction Λ which assigns to a given rank r oriented matroid M on n points a rank n + r oriented matroid Λ(M) on 2n points such that the face lattice of Λ(M) is polytopal if and only if M is realizable. The Λ-construction generalized a technique used by Perles to construct a nonrational polytope [10]. It was used by Lawrence to prove that the class of polytopal lattices is strictly contained in the class of face lattices of oriented matroids (unpublished) and by Billera and Munson to show that the latter class is not closed under polarity. See [4] for a discussion of this construction and both of these applications.