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