Prodsimplicial neighborly polytopes

Benjamin Matschke (1) , Julian Pfeifle (2) , Vincent Pilaud (3)

1 IHES - Institut des Hautes Études Scientifiques
2 UPC - Universitat Politècnica de Catalunya = Université polytechnique de Catalogne [Barcelona]
3 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their $k$-skeleton is combinatorially equivalent to that of a product of $r$ simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general $r$ and $k$, the lowest dimension we achieve is $2k + r + 1$. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of $r$ simplices, when the dimensions of these simplices are all large compared to $k$.

Mots clés: Neighborly polytope; Product of simplices; Skeleta preserving projection

Domaines: Combinatoire [math.CO]

Benjamin Matschke, Julian Pfeifle, Vincent Pilaud. Prodsimplicial neighborly polytopes. Discrete and Computational Geometry, 2011, 46 (1), pp.100-131. ⟨10.1007/s00454-010-9311-y⟩. ⟨hal-00777936⟩