Many neighborly polytopes and oriented matroids

Details Many neighborly polytopes and oriented matroids Many neighborly polytopes and oriented matroids

2012-02-13

arxiv.org/abs/1202.2810v1

Mathematics

Metric Geometry

24 pages, 3 figures

Scientific paper

We say that a d-polytope P is neighborly if every subset of at most d/2 vertices is a face of P. This concept applies naturally to oriented matroids. In 1982, Shemer introduced a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. We generalize this construction and extend it to oriented matroids. Moreover we provide a new construction that allows to extend balanced oriented matroids. In the dual setting, this means that given a neighborly matroid of rank d with n elements, we obtain a new neighborly matroid of rank d+1 with n+1 elements. With the help of this new construction, we are able to generate many non-realizable neighborly matroids and to prove that the number of neighborly d-polytopes with n vertices is greater than ((n-1)/e^3/2)^d(n-1)/2 when n>2d.

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