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Unformatted text preview: C&O 355: Lecture 17 Notes Nicholas Harvey http://www.math.uwaterloo.ca/~harvey/ 1 Faces of Polyhedra Definition 1.1. An affine space A R n is any set of the form A = { x + z : x L } , where L R n is a linear subspace and z R n is any vector. The dimension of A , denoted dim A , is simply the dimension of the corresponding linear subspace L . Definition 1.2. Let C R n be any set. The dimension of C , denoted dim C , is min { dim A : C A } , where the minimum is taken over all affine spaces A . Strictly speaking, if C = then dim C is not defined. We will adopt the convention that dim C = 1. Definition 1.3. Let C R n be any convex set. An inequality a T x b is called a valid inequality if a T x b holds for every point x C . Definition 1.4. Let P R n be a polyhedron. A face of P is any set F of the form F = P n x R n : a T x = b o , (1.1) where a T x b is a valid inequality for P . Notice that any face of a polyhedron is itself a polyhedron. As an example, if we take a = and b = 0 then this shows that P is itself a face. On the other hand, if we take a = and b = 1 then this shows that is a face. Definition 1.5. Let P be a polyhedron and let F P be a face. Then F is called a d dimensional face or a dface if dim F = d . Note that if F consists of a single point v (i.e., F = { v } ) then F is a 0face. In this case, v is the unique maximizer of a over P , where a is the vector in Eq. (1.1), and thus v is a vertex of P . Recalling our earlier results, we see that for polyhedra, vertices, extreme points, basic feasible solutions, and 1faces are the same concept. Definition 1.6. If dim P = d and F P is a ( d 1)face then F is called a facet . Definition 1.7. A 1face of a polyhedron is called an edge . 1 Fact 1.8. Let P be a polytope with dim P = d and let F be a face of P . Then F is itself a polytope and so it too has faces. The faces of F are precisely the faces of P that are contained in F . Now assume that F is a facet. The facets of F are precisely the ( d 2)faces of P that are contained in F . Furthermore, each facet of F can be obtained as the intersection of F and another facet of P ....
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 Fall '09
 Harvey

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