In their paper proving the Hirsch bound for flag normal simplicial complexes(Math. Oper.~Res.~2014) Adiprasito and Benedetti define the notionof~\emph{combinatorial segment}. The study of the maximal length of theseobjects provides the upper bound~$O(n2^d)$ for the diameter of any normal puresimplicial complex of dimension~$d$ with~$n$ vertices, and the Hirsch bound$n-d$ if the complexes are, moreover, flag. In the present article, we proposea formulation of combinatorial segments which is equivalent but more local, byintroducing the notions of monotonicity and conservativeness of dual paths inpure simplicial complexes. We use this definition to investigate furtherproperties of combinatorial segments. Besides recovering the two stated bounds,we show a refined bound for banner complexes, and study the behavior of themaximal length of combinatorial segments with respect to two usual operations,namely join and one-point suspension. Finally, we show the limitations ofcombinatorial segments by constructing pure normal simplicial complexes inwhich all combinatorial segments between two particular facets achieve thelength $\Omega(n2^{d})$. This includes vertex-decomposable---thereforeHirsch---polytopes.
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