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A comparison theorem for $f$-vectors of simplicial polytopes | | Anders Björner
; | | Date: |
12 May 2006 | | Subject: | Combinatorics | | Abstract: | Let $S(n,d)$ and $C(n,d)$ denote, respectively, the stacked and the cyclic $d$-dimensional polytopes on $n$ vertices. Furthermore, $f_i(P)$ denotes the number of $i$-dimensional faces of a polytope $P$.
Theorem. Let $P$ be a $d$-dimensional simplicial polytope. Suppose that $$f_r(S(n_1,d))le f_r(P) le f_r(C(n_2,d))$$ for some integers $n_1, n_2$ and $rle d-2$. Then, $$f_s(S(n_1,d))le f_s(P) le f_s(C(n_2,d))$$ for all $s$ such that $r<s<d$.
Some special cases were previously known. For $r=0$ these inequalities are the well-known lower and upper bound theorems for simplicial polytopes. The $s=d-1$ case of the upper bound part is contained in the ``generalized upper bound theorem’’ of Kalai.
The result is implied by a more general ``comparison theorem’’ for $f$-vectors, formulated in Section 1. Among its other consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes. | | Source: | arXiv, math/0605336 | | Services: | Forum | Review | PDF | Favorites | |
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