Abstract

. We start with a theorem of Perles on the k-skeleton, Skel k (P ) (faces of dimension k) of d- polytopes P with d+b vertices for large d. The theorem says that for fixed b and d, if d is sufficiently large, then Skel k (P ) is the k-skeleton of a pyramid over a (d \Gamma 1)-dimensional polytope. Therefore the number of combinatorially distinct k-skeleta of d-polytopes with d + b vertices is bounded by a function of k and b alone. Next we replace b (the number of vertices minus the dimension) by related but deeper invariants of P , the g-numbers. For a d-polytope P there are [d=2] invariants g1 (P ); g2 (P ); :::; g [d=2] (P ) which are of great importance in the combinatorial theory of polytopes. We study polytopes for which g k is small and carried away to related and slightly related problems. Key words: Convex polytopes, skeleton, simplicial sphere, simplicial manifold, f-vector, g- theorem, ranked atomic lattices, stress, rigidity, sunflower, lower bound theorem, elementary poly...

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