The g - Theorem MA 715 Course Notes Spring 2002

These notes will be rather informal in many places. For more precision, refer to Lectures on Polytopes by Günter Ziegler [Zie95], Convex Polytopes by Branko Grünbaum [Grü67], An Introduction to Convex Polytopes by Arne Brøndsted [Brø83], and Convex Polytopes and the Upper Bound Conjecture by Peter McMullen and Geoffrey Shephard [MS71]. See also the handbook papers [BL93, KK95] We will be studying polyhedra and polytopes. A (convex) polyhedron is the intersection of a finite number of closed halfspaces in R. A (convex) polytope is a bounded polyhedron; equivalently, it is the convex hull (the smallest convex set containing) a given finite set of points. If P is a polyhedron and H is a hyperplane such that P is contained in one of the closed halfspaces associated with H and F := H∩P is nonempty, then H is a supporting hyperplane of P . In such a case, if F 6= P , then F is a proper face of P . The improper faces of P are the empty set and P itself. Let F(P ) denote the set of all faces of P , both proper and improper, and let F(bdP ) := F(P ) \ {P}. The dimension of a subset of R is the dimension of its affine span. If P is a polyhedron and dimP = d, then P is called a d-polyhedron, and faces of P of dimension 0, 1, d − 2, and d − 1 are called vertices, edges, subfacets (or ridges), and facets of P , respectively. We denote the number of j-dimensional faces (j-faces) of P by fj(P ) (or simply fj when the polyhedron is clear) and call f(P ) := (f0(P ), f1(P ), . . . , fd−1(P )) the f -vector of P . The empty set is the unique face of dimension −1 and P is the unique face of dimension d, so f−1(P ) = 1 and fd(P ) = 1. The big problem is to understand/describe f(P) := {f(P ) : P is a d-polytope}!