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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].

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