Partition problems in discrete geometry

This Thesis deals with the following type of problems, which we denote partition problems, Given a set X in IR, is there a way to partition X such that the convex hulls of all parts satisfy certain combinatorial properties? We focus on the following two kinds of partition problems. • Tverberg type partitions. In this setting, one of the properties we ask the sets to satisfy is that their convex hulls all intersect. • Ham sandwich type partitions. In this setting, one of the properties we ask the sets to satisfy is that the interior of their convex hulls are pairwise disjoint. The names for these types of partitions come from the quintessential theorem from each type, namely Tverberg’s theorem and the ham sandwich theorem. We present a generalisation and a variation of each of these classic results. The generalisation of the ham sandwich theorem extends the classic result to partitions into any arbitrary number of parts. This is presented in chapter 2. Then, in chapter 3, variations of the ham sandwich Theorem are studied when we search for partitions such that every hyperplane avoids an arbitrary number of sections. These results appear in two papers, [39, 35]. The generalisation of Tverberg’s theorem consists of adding a condition of tolerance to the partition. Namely, that we may remove an arbitrary number of points and the partition still is Tverberg type. This is presented in chapter 4. Then, in chapter 5, “colourful” variations of Tverberg’s Theorem are studied along their applications to some purely combinatorial problems. These results appear in two papers, [41, 40].