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Let $\Delta(d,n)$ denote the maximum diameter of a $d$-dimensional polyhedronwith $n$ facets. In this paper, we propose a unified analysis of a recursiveinequality about $\Delta(d,n)$ established by Kalai and Kleitman in 1992. Thisyields much simpler proofs of a tail--polynomial and tail--almost--linearbounds on $\Delta(d,n)$ which are recently discussed by Gallagher and Kim.

A simple proof of tail--polynomial bounds on the diameter of polyhedra