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## 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