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Cyclic Polytopes and Oriented Matroids

✍ Scribed by Raul Cordovil; Pierre Duchet

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 244 KB
Volume: 21
Category: Article
ISSN: 0195-6698
DOI: 10.1006/eujc.1999.0317

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✦ Synopsis

Consider the moment curve in the real euclidean space R d defined parametrically by the map γ : R → R d , t → γ (t) = (t, t 2 , . . . , t d ). The cyclic d-polytope C d (t 1 , . . . , t n ) is the convex hull of n > d different points on this curve. The matroidal analogs are the alternating oriented uniform matroids. A polytope (resp. matroid polytope) is called cyclic if its face lattice is isomorphic to that of C d (t 1 , . . . , t n ). We give combinatorial and geometrical characterizations of cyclic (matroid) polytopes. A simple evenness criterion determining the facets of C d (t 1 , . . . , t n ) was given by Gale [21]. We characterize the admissible orderings of the vertices of the cyclic polytope, i.e., those linear orderings of the vertices for which Gale's evenness criterion holds. Proofs give a systematic account on an oriented matroid approach to cyclic polytopes.

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