Mathematical Sciences: Combinatorics of Convex Polytopes

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9410679 Chan It is proposed to investigate combinatorial properties of convex polytopes and more general polyhedral complexes which can be decomposed in an orderly fashion. Such complexes are called shellable. Shellability has proved to be an important tool in the combinatorics of convex polytopes. The focus will be on a combinatorial invariant called the toric h-vector, which in certain cases is simply a fixed linear transformation of the vector of face-numbers of the given complex. Until recently the only known interpretation of the toric h-vector of an arbitrary rational polytope came from toric variety theory, but recently McMullen discovered a combinatorial meaning involving the polytope algebra, in the case of simplicial polytopes. McMullen's famous g-theorem says that the toric hvectors of simplicial polytopes are characterized by the Macaulay inequalities. An attempt will be made to determine whether the same (or stronger) inequalities hold for toric h-vectors of other shellable complexes. Aside from shelling techniques, it might be possible to use the polytope algebra, which will also be studied. Of special interest to the investigator and collaborators will be zonotopes, which are orthogonal projections of higher dimensional cubes. ***