- 100 Downloads
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 & 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.
KeywordsNeighborly polytope Product of simplices Skeleta preserving projection
- 1.Amenta, N., Ziegler, G.M.: Deformed products and maximal shadows of polytopes. In: Advances in Discrete and Computational Geometry. Contemporary Mathematics, vol. 223, pp. 57–90. Am. Math. Soc., Providence (1999) Google Scholar
- 5.Kneser, M.: Aufgabe 360. Jahresber. Dtsch. Math.-Ver. 2 58, 27 (1955) Google Scholar
- 8.Matoušek, J.: Using the Borsuk–Ulam theorem. In: Universitext. Lectures on Topological Methods in Combinatorics and Geometry. Springer, Berlin (2003). Written in cooperation with A. Björner and G.M. Ziegler Google Scholar
- 9.Matschke, B., González Merino, B., Pfeifle, J., Pilaud, V.: Prodsimplicial neighborly polytopes. In: Noy, M., Pfeifle, J. (eds.) i-Math Winter School DocCourse Discrete and Computational Geometry. Research Reports, vol. III, pp. 105–139. Centre de Recerca Matemàtica, Barcelona (2009) Google Scholar
- 10.Rörig, T., Sanyal, R.: Non-projectability of polytope skeleta (2009). arXiv:0908.0845, 18 pages