eprintid: 1398297 rev_number: 34 eprint_status: archive userid: 608 dir: disk0/01/39/82/97 datestamp: 2014-01-24 16:48:29 lastmod: 2019-10-19 08:26:11 status_changed: 2014-01-24 16:48:29 type: thesis metadata_visibility: show item_issues_count: 0 creators_name: Soberon Bravo, P title: Partition problems in discrete geometry ispublished: unpub divisions: A01 divisions: B04 divisions: C06 keywords: Discrete geometry, Combinatorial geometry, Tverberg's theorem, Convexity, Partitions of measures abstract: This thesis deals with the following type of problems, which we denote partition problems, Given a set X in R^d, 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. 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. date: 2013-07-28 vfaculties: VMPS oa_status: green full_text_type: other thesis_class: doctoral_open language: eng thesis_view: UCL_Thesis dart: DART-Europe primo: open primo_central: open_green verified: verified_manual elements_source: Manually entered elements_id: 881596 lyricists_name: Soberon Bravo, Pablo lyricists_id: PSOBE40 full_text_status: public pagerange: 1 - 86 pages: 86 institution: UCL (University College London) department: Mathematics thesis_type: Doctoral editors_name: Bárány, I editors_name: Ball, K citation: Soberon Bravo, P; (2013) Partition problems in discrete geometry. Doctoral thesis , UCL (University College London). Green open access document_url: https://discovery.ucl.ac.uk/id/eprint/1398297/1/soberon-thesis-final.pdf