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Some aspects of the combinatorial theory of convex polytopes. (English) Zbl 0804.52006
Bisztriczky, T. (ed.) et al., Polytopes: abstract, convex and computational. Proceedings of the NATO Advanced Study Institute, Scarborough, Ontario, Canada, August 20 - September 3, 1993. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 440, 205-229 (1994).
Summary: We start with a theorem of Perles on the $$k$$-skeleton, $$\text{Skel}_ k(P)$$ (faces of dimension $$\leq k$$) 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 $$\text{Skel}_ k(P)$$ is the $$k$$-skeleton of a pyramid over a $$(d -1)$$-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 $$g_ 1(P), g_ 2(P),\dots,g_{[d/2]}(P)$$ which are of great importance in the combinatorial theory of polytopes. We study polytopes for which $$g_ k$$ is small and move to related and slightly related problems.
For the entire collection see [Zbl 0797.00016].

MSC:
 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)