A comparison theorem for $f$-vectors of simplicial polytopes

Details A comparison theorem for $f$-vectors of simplicial polytopes A comparison theorem for $f$-vectors of simplicial polytopes

2006-05-12

arxiv.org/abs/math/0605336v4

Mathematics

Combinatorics

8 pages. Revised and corrected version. To appear in "Pure and Applied Mathematics Quarterly"

Scientific paper

Let $f_i(P)$ denote the number of $i$-dimensional faces of a convex polytope $P$. Furthermore, let $S(n,d)$ and $C(n,d)$ denote, respectively, the stacked and the cyclic $d$-dimensional polytopes on $n$ vertices. Our main result is that for every simplicial $d$-polytope $P$, if $$ f_r(S(n_1,d))\le f_r(P) \le f_r(C(n_2,d)) $$ for some integers $n_1, n_2$ and $r$, then $$ f_s(S(n_1,d))\le f_s(P) \le f_s(C(n_2,d)) $$ for all $s$ such that $r

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