Mathematical Sciences: Valuations and the Combinatorics of Convex Sets

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This research will apply the properties of the valuation module of the lattice of finite unions of convex polytopes to the study of combinatorial problems concerning convex sets. A vehicle for this attack will be the lattices generated from finite subsets X of Euclidean space by taking unions and intersections of the convex hulls of subsets of X. A wealth of such combinatorial problems exists. Those closely related to Tverberg's theorem will be singled out, for example a conjecture of John Reay, as being particularly suited to the methods of our attack. This research will consider lattices of finite unions of convex polytopes. A polytope in space is a region with flat sides. It is convex if the segment joining any two points in the region lies in the region. Starting with a finite set of points and joining them together and taking unions of the resulting regions forms such a lattice and makes for an important combinatorial and geometrical object of study. This proposal will bring to bear novel techniques for obtaining properties of these lattices. It should bear interesting fruit.