A universality theorem for projectively unique polytopes and a conjecture of Shephard
We prove that every polytope described by algebraic coordinates is the face of a projectively unique polytope. This provides a universality property for projectively unique polytopes. Using a closely related result of Below, we construct a combinatorial type of 5-dimensional polytope that is not realizable as a subpolytope of any stacked polytope. This disproves a classical conjecture in polytope theory, first formulated by Shephard in the seventies.
KeywordsUnique Point Projective Transformation Combinatorial Type Projective Basis Oriented Matroid
Unable to display preview. Download preview PDF.
- A. Below, Complexity of triangulation, Ph.D. thesis, ETH Zürich, Zürich, CH, 2002.Google Scholar
- M. Dobbins, Representations of polytopes, Ph.D. thesis, Temple University, Philadelphia, PA, 2011.Google Scholar
- G. Kalai, Polytope skeletons and paths, in Handbook of Discrete and Computational Geometry, Chapman & Hall/CRC, Boca Raton, FL, 2004, pp. 455–476.Google Scholar
- G. Kalai, Open problems for convex polytopes I’d love to see solved, Talk on Workshop ”Convex Polytopes” at RIMS Kyoto, July 2012, slides available at http://www.gilkalai.files.wordpress.com/2012/08/kyoto-3.pdf.
- K. G. C. von Staudt, Beiträge zur Geometrie der Lage, no. 2, Baur und Raspe, Nürnberg, 1857.Google Scholar