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'polytope'
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| Title of object: |
polytope |
| Canonical Name: |
Polytope |
| Type: |
Definition |
| Created on: |
2004-02-03 22:51:48 |
| Modified on: |
2004-02-13 15:33:21 |
| Classification: |
msc:52B40 |
| Keywords: |
polytope, face lattice, graded poset |
| Defines: |
face lattice, f-vector, flag f-vector |
Revision comment (for changes between this and next version):
| Usually the $f$-vector does not include the (improper) $d$-dimensional face. |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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Content:
\PMlinkescapeword{satisfy}
\PMlinkescapeword{inclusion}
\PMlinkescapeword{collection}
\PMlinkescapeword{dimension}
\PMlinkescapeword{dimensions}
\PMlinkescapeword{spheres}
\PMlinkescapeword{index}
\PMlinkescapeword{relation}
\PMlinkescapeword{relations}
\PMlinkescapeword{identities}
\PMlinkescapeword{meet}
\PMlinkescapeword{meets}
\PMlinkescapeword{join}
\PMlinkescapeword{lattice}
\PMlinkescapeword{extension}
\PMlinkescapeword{formula}
A convex \emph{polytope} $P$ is the convex hull of a finite set of points
in $\mathbb{R}^n$, or alternatively a \PMlinkname{bounded}{Bounded} intersection
of finitely many halfspaces of $\mathbb{R}^n$.
The dimension of $P$ is the smallest $d$ such that $P$ can be embedded in
$\mathbb{R}^d$. A $d$-dimensional polytope is also called a $d$-polytope.
An affine hyperplane $H$ is a supporting hyperplane of $P$ if it has a nonempty
intersection with $P$ and further $P$ is fully contained in one of the two halfspaces defined by $H$. In this situation, the intersection $P\cap H$ is called a face of $P$ and is itself a polytope. It is called a facet if it has the same dimension as the hyperplane. Some authors also require the empty set to be a $-1$-dimensional face of every polytope. The polytope $P$ is usually taken to be an improper face of itself.
The notation $f_i$ is used to denote the number of $i$-dimensional faces
of $P$; thus $f_{-1}=1$ and $f_d=1$.
The vector $(f_0, f_1,\dots, f_d)$ is called the $f$-vector of $P$.
For example, if $P$ is the 3-dimensional cube, then the $f$-vector of
$P$ is $(8,12,6)$.
The entries in the $f$-vector satisfy the Euler-Poincar\'{e} formula
(sometimes just called the Poincar\'{e} formula):
\begin{equation*}
\sum_{-1\le i\le d}(-1)^i f_i=0.
\end{equation*}
The face \PMlinkname{lattice}{Lattice} $\mathcal{L}(P)$ of a polytope $P$ consists of all the
faces of $P$, ordered by inclusion; the meet of two faces is their intersection,
and the join of two faces is the face of minimal dimension containing their
union. This lattice is graded and Eulerian.
For any graded poset with maximum and minimum elements there is an extension
of the $f$-vector called the flag $f$-vector. For any subset $S$ of
$\{0,1,\dots,d - 1\}$, the $f_S$ entry of the flag $f$-vector of $P$
is the number of chains of faces in $\mathcal{L}(P)$ with dimensions
coming only from $S$.
The flag $f$-vector of a three-dimensional cube is given in the following
table:
\begin{center}
\begin{tabular}{cl}
$S$ & $f_S$ \\
\hline
$\emptyset$ & 1 \\
$\{0\}$ & 8 \\
$\{1\}$ & 12 \\
$\{2\}$ & 6 \\
$\{0,1\}$ & 8*3=24 \\
$\{0,2\}$ & 8*3=24 \\
$\{1,2\}$ & 12*2=24 \\
$\{0,1,2\}$ & 8*3*2=48
\end{tabular}
\end{center}
For example, $f_{\{1,2\}}=24$ because each of the 12 edges
meets exactly two faces.
Although the flag $f$-vector of a $d$-polytope has $2^d$ entries, most
of them are redundant, as they satisfy a collection of identities
generalizing the
Euler-Poincar\'{e} formula and called the generalized Dehn-Sommerville
relations. Interestingly, the number of nonredundant entries in the
flag $f$-vector of a $d$-polytope is the Fibonacci
number $F_{d-1}$.
\begin{thebibliography}{3}
\bibitem{cite:BB}
Bayer, M. and L. Billera, \emph{Generalized Dehn-Sommerville relations for
polytopes, spheres and Eulerian partially ordered sets}, Invent. Math. 79
(1985), no. 1, 143--157.
\bibitem{cite:BK}
Bayer, M. and A. Klapper, \emph{A new index for polytopes}, Discrete Comput.
Geom. 6
(1991), no. 1, 33--47.
\bibitem{cite:Z}
Ziegler, G., \emph{Lectures on polytopes}, Springer-Verlag, 1997.
\end{thebibliography} |
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