polytope
A polytope^{} is the convex hull^{} of finitely many points in Euclidean space. A polytope constructed in this way is the convex hull of its vertices and is called a $\mathrm{V}$-polytope. An $\mathrm{H}$-polytope is a bounded^{} intersection^{} of upper halfspaces. By the WeylâMinkowski theorem, these descriptions are equivalent^{}, that is, every $\mathrm{\u0111\x9d\x92\pm}$-polytope is an $\mathrm{\xe2\x84\x8b}$-polytope, and vice versa. This shows that our intuition, based on the study of low-dimensional polytopes, that one can describe a polytope either by its vertices or by its facets is essentially correct.
The dimension^{} of $P$ is the smallest $d$ such that $P$ can be embedded in ${\mathrm{\xe2\x84\x9d}}^{d}$. A $d$-dimensional polytope is also called a $d$-polytope.
A face of a polytope is the intersection of the polytope with a supporting hyperplane. Intuitively, a supporting hyperplane is a hyperplane^{} that âjust touchesâ the polytope, as though the polytope were just about to pass through the hyperplane. Note that this intuitive picture does not cover the case of the empty face, where the supporting hyperplane does not touch the polytope at all, or the fact that a polytope is a face of itself. The faces of a polytope, when partially ordered by set inclusion, form a geometric lattice, called the face lattice^{} of the polytope.
The Euler polyhedron formula^{}, which states that if a $3$-polytope has $V$ vertices, $E$ edges, and $F$ faces, then
$$V-E+F=2,$$ |
has a generalization^{} to all $d$-polytopes. Let $({f}_{-1}=1,{f}_{0},\mathrm{\xe2\x80\u0160},{f}_{d-1},{f}_{d}=1)$ be the f-vector of a $d$-polytope $P$, so ${f}_{i}$ is the number of $i$-dimensional faces of $P$. Then these numbers satsify the EulerâPoincarĂ©âSchlĂ€fli formula:
$$\underset{i=-1}{\overset{d}{\xe2\x88\x91}}{(-1)}^{i}\xe2\x81\u0105{f}_{i}=0.$$ | (1) |
This is the first of many relations^{} among entries of the f-vector satisfied by all polytopes. These relations are called the DehnâSommerville relations. Any poset which satisfies these relations is Eulerian (http://planetmath.org/EulerianPoset), so the face lattice of any polytope is Eulerian.
References
- 1 Bayer, M. and L. Billera, Generalized DehnâSommerville relations for polytopes, spheres and Eulerian partially ordered sets^{}, Invent. Math. 79 (1985), no. 1, 143â157.
- 2 Bayer, M. and A. Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), no. 1, 33â47.
- 3 Minkowski, H. Allgemeine LehrsĂ€tze ĂŒber die konvexe Polyeder, Nachr.Â Ges.Â Wiss., GĂ¶ttingen, 1897, 198â219.
- 4 Weyl, H. Elementare Theorie der konvexen Polyeder, Comment.Â Math.Â Helvetici, 1935, 7
- 5 Ziegler, G., Lectures on polytopes, Springer-Verlag, 1997.
Title | polytope |
Canonical name | Polytope |
Date of creation | 2013-03-22 14:07:59 |
Last modified on | 2013-03-22 14:07:59 |
Owner | mps (409) |
Last modified by | mps (409) |
Numerical id | 26 |
Author | mps (409) |
Entry type | Definition |
Classification | msc 52B40 |
Related topic | Polyhedron |
Related topic | PoincareFormula |
Related topic | EulersPolyhedronTheorem |
Defines | V-polytope |
Defines | H-polytope |
Defines | d-polytope |
Defines | dimension |