Combinatorial Aspects of Convex Polytopes

This chapter discusses some techniques that have been successful in analyzing the combinatorial aspects of convex polytopes. A convex polyhedron is a subset of ℝ d that is the intersection of a finite number of closed half-spaces. A bounded convex polyhedron is called a convex polytope. The following can be regarded as the fundamental theorem of convex polytopes: P ⊂ ℝ d is a polytope if and only if it is the convex hull of a finite set of points in ℝ d . This theorem and related results are foundational to the theory of linear programming duality, and one of the central themes of combinatorial optimization is to make this conversion for special polytopes related to specific programming problems. The problem of developing algorithms to convert from one description of a polytope to the other arises in mathematical programming and computational geometry. The second theorem states that the collection of all the faces of a polyhedron P , ordered by inclusion, is a lattice. This lattice is called the face lattice or boundary complex of P , and two polytopes are (combinatorially) equivalent if their face lattices are isomorphic. The third theorem states that the face lattices of P and P * are anti-isomorphic. Two polytopes with anti-isomorphic face lattices are said to be dual. Two important dual classes of d -polytopes are the class of simplicial d -polytopes and the class of simple d -polytopes.

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