Volume 114, 1990
Some Recent Results on Convex Polytopes
CARL W. LEE
ABSTRACT. We sample a few results on the combinatorial structure of convex
polytopes, including Lawrence's volume formula, f-vectors and h-vectors,
associated algebraic structures, shellability, bistellar operations and p.1.-
spheres, connections with stress and rigidity, triangulations, winding num-
bers, the moment map, and canonical convex combinations.
The study of polyhedra has enjoyed rapid growth, stimulated partly by
the development of mathematical programming in the last few decades, and
partly by more recently discovered connections with commutative algebra
and algebraic geometry. We informally survey a few results on the combina-
torial structure of convex polytopes, beginning with Lawrence's volume for-
mula. This leads naturally into the notions of the f-vector and the h-vector.
These, in turn, have algebraic significance in associated algebraic structures.
Examining these structures in the context of two inductive methods for con-
structing polytopes, shellings and bistellar operations, reveals an interplay
with stress and rigidity. One consequence is a new proof that p.1.-spheres are
Cohen-Macaulay. Gale transforms play a role here and can be used to define
a class of triangulations of a convex polytope. They also provide a geometric
interpretation of the h-vector in terms of winding numbers. We conclude
with a brief discussion of a toric variety associated with a rational simplicia1
convex polytope. The components of the h-vector appear as the dimensions
of its homology groups, and its moment map suggests a canonical way to
express a point of the polytope as a convex combination of the vertices.
2. Lawrence's volume formula
Let us start by considering a d-dimensional convex polyhedron P of the
Ax 5 b ,
2 0 ) , where A is an rn x d matrix and
Mathematics Subject Classijication
Supported, in part, by grant DMS-8802933 of the National Science Foundation.
1990 American Mathematical Society
$.25 per page