Abstract
The theory of oriented matroids is applied to the class of neighborly convex polytopes. After giving shortened and purely combinatorial proofs for various known properties of cyclic and neighborly polytopes, we focus our attention on a very interesting property of neighborly chirotopes. We establish the combinatorial analogue to a theorem of I. Shemer: The chirotope of a neighborly 2k-polytope P is rigid, i.e. the entire internal structure of P is uniquely determined by its boundary complex.As the main new result we give a negative answer to a question of M. A. Perles: The property to be rigid does not characterize the neighborly 2k-polytopes among all simplicial polytopes.
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