A polytopeMathworldPlanetmath is the convex hullMathworldPlanetmath of finitely many points in Euclidean space. A polytope constructed in this way is the convex hull of its vertices and is called a V-polytope. An H-polytope is a boundedPlanetmathPlanetmath intersectionMathworldPlanetmath of upper halfspaces. By the Weyl–Minkowski theorem, these descriptions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath, that is, every đ’±-polytope is an ℋ-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 dimensionPlanetmathPlanetmath of P is the smallest d such that P can be embedded in ℝd. 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 hyperplaneMathworldPlanetmathPlanetmath that “just touches” 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 latticeMathworldPlanetmath of the polytope.

The Euler polyhedron formulaMathworldPlanetmathPlanetmath, which states that if a 3-polytope has V vertices, E edges, and F faces, then


has a generalizationPlanetmathPlanetmath to all d-polytopes. Let (f-1=1,f0,
,fd-1,fd=1) be the f-vector of a d-polytope P, so fi is the number of i-dimensional faces of P. Then these numbers satsify the Euler–Poincaré–SchlĂ€fli formula:

∑i=-1d(-1)iⁱfi=0. (1)

This is the first of many relationsMathworldPlanetmath among entries of the f-vector satisfied by all polytopes. These relations are called the Dehn–Sommerville relations. Any poset which satisfies these relations is Eulerian (http://planetmath.org/EulerianPoset), so the face lattice of any polytope is Eulerian.


  • 1 Bayer, M. and L. Billera, Generalized Dehn–Sommerville relations for polytopes, spheres and Eulerian partially ordered setsMathworldPlanetmath, Invent. Math. 79 (1985), no. 1, 143–157.
  • 2 Bayer, M. and A. Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), no. 1, 33–47.
  • 3 Minkowski, H. Allgemeine LehrsĂ€tze ĂŒber die konvexe Polyeder, Nachr. Ges. Wiss., Göttingen, 1897, 198–219.
  • 4 Weyl, H. Elementare Theorie der konvexen Polyeder, Comment. Math. 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