** Next:** What is the face
** Up:** Convex Polyhedron
** Previous:** What is convex polytope/polyhedron?
** Contents**

##

What are the faces of a convex polytope/polyhedron?

Let be a convex -polyhedron (or -polytope) in .

For a real -vector and a real number , a linear inequality
is called *valid* for if
holds for all .
A subset of a polyhedron is called a *face* of if it is
represented as

for some valid inequality
. By this definition,
both the empty set and the whole set are
faces. These two faces are called *improper* faces while the other
faces are called *proper* faces.
We can define faces geometrically. For this, we need to
define the notion of supporting hyperplanes.
A hyperplane of is *supporting
* if one of the two closed halfspaces of contains .
A subset of is called a *face* of
if it is either ,
itself or the intersection of with a supporting hyperplane.

The faces of dimension 0, ,
and
are called the *vertices*,
*edges*, *ridges* and *facets*, respectively.
The vertices coincide
with the *extreme points* of which are defined as points which cannot
be represented as convex combinations of two other points in .
When an edge is not bounded, there are two cases: either it is a line
or a half-line starting from a vertex.
A half-line edge is called an *extreme ray*.

** Next:** What is the face
** Up:** Convex Polyhedron
** Previous:** What is convex polytope/polyhedron?
** Contents**
Komei Fukuda
2004-08-26