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In 1992, Kalai and Kleitman proved a quasipolynomial upper bound on thediameters of convex polyhedra. Todd and Sukegawa-Kitahara provedtail-quasipolynomial bounds on the diameters of polyhedra. These tail boundsapply when the number of facets is greater than a certain function of thedimension. We prove tail-quasipolynomial bounds on the diameters of polytopesand normal simplicial complexes. We also prove tail-polynomial upper bounds onthe diameters of polyhedra.

Tail diameter upper bounds for polytopes and polyhedra