A comparison theorem for $f$-vectors of simplicial polytopes
| dc.creator | Björner, Anders | |
| dc.date | 2006-05-12 | |
| dc.date | 2006-11-07 | |
| dc.date.accessioned | 2026-07-07T07:14:10Z | |
| dc.date.available | 2026-07-07T07:14:10Z | |
| dc.description | 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 $r<s$. For $r=0$ these inequalities are the well-known lower and upper bound theorems for simplicial polytopes. The result is implied by a certain ``comparison theorem'' for $f$-vectors, formulated in Section 4. Among its other consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes. | |
| dc.description | 8 pages. Revised and corrected version. To appear in "Pure and Applied Mathematics Quarterly" | |
| dc.identifier | https://arxiv.org/abs/math/0605336 | |
| dc.identifier | http://arxiv.org/abs/math/0605336 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/112766 | |
| dc.subject | Combinatorics | |
| dc.subject | 05E99; 52B05 | |
| dc.title | A comparison theorem for $f$-vectors of simplicial polytopes | |
| dc.type | text |