By Gyula Karolyi

Abstract. Polytopes (bounded polyhedra, or convex hulls of finite set of points) are natural objects of study in combinatorics, geometry and optimization.

Every polytope can be expressed as a finite union of simplices, which are, in some sense, the simplest polytopes. One may even require that the vertices of these simplices are vertices of the polytope itself. Information about the polytope then can be gained through the study of these simple building units. For example, the area of a polygon with n vertices is simply the sum of the areas of n-2 triangles.

The central theme of this talk is an exclusion-inclusion formula to express a polytope as a "signed union" of simplices, where, instead of vertices, we fix the supporting hyperplanes of the simplices. This kind of representation is useful if the polytope is given as the intesection of half-spaces. For example, every quadrilateral, save the parallelograms, can be obtained by removing a triangle from a larger triangle determined by three appropriate sides of the polygon.

Such representation theorems were obtained independently by Varchenko, Lawrence, Karolyi-Lovasz, and Filliman, and motivations range from volume computation through discrepancy theory to the study of general hypergeometric functions. Filliman's result is extremally nice as it establishes the desired representation through a triangulation of "the" dual polytope.

We also try to explore the connection of these results with classical angle-sum relations for polytopes (Gram's theorem) and hyperplane arrangements.