Mathematical Sciences: Valuations and the Combinatorics of Convex Sets

本研究将凸多面体有限并格的赋值模的性质应用于研究与凸集有关的组合问题。这种攻击的一个工具是从欧几里得空间的有限子集X通过取X的子集的凸包的并集和交集生成的格。存在大量这样的组合问题。那些与特沃伯格定理密切相关的，例如约翰雷伊的一个猜想，将被挑选出来，因为它特别适合于我们的攻击方法。本研究将考虑凸多面体的有限并格。空间中的多面体是具有平坦边的区域。如果连接区域中任意两点的线段位于该区域内，则该区域是凸的。从一组有限的点开始，将它们连接在一起，并将所产生的区域合并，形成这样一个网格，并成为一个重要的组合和几何研究对象。这一建议将带来承担新的技术来获得这些晶格的属性。它会结出有趣的果实。

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.