A new basis of polytopes

Abstract Let P be a d-polytope. For a set of indices S = {1, i2, …, ik}, 0 ⩽ i1 A remarkable recent theorem by Bayer and Billera asserts that the dimension of the real affine space spanned by flag vectors of d-polytopes is cd − 1, where cd is the dth Fibonacci number. We will construct a new family of d-polytopes whose flag vectors affinely span this space. As a consequence, we compute the dimensions of the affine span of flag vectors of several subclasses of d-polytopes. We show how the structure of intervals in face lattices of polytopes affects their flag numbers. Our proofs are by appropriate computations in the incidence algebra of face lattices of polytopes. We study also linear inequalities for the flag numbers and face numbers of d-polytopes, and discuss some connections with the new notion of h-vectors for arbitrary polytopes ( R. Stanley, “Enumerative Combinatorics”, Vol. I, Chap. 4, Wadsworth, Monterey, CA, 1986 , and Generalized h-vectors, intersection cohomology of toric varieties, and related results, in “Proceedings, Japan-USA Workshop on Commutative Algebra and Combinatorics,” to appear).

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