Roldán Pensado, E;
(2013)
Problems in Convex Geometry.
Doctoral thesis , UCL (University College London).

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Abstract
We deal with five different problems from convex geometry, each on its own chapter of this Thesis. These problems are the following. Random copies of a convex body: We study the probability that a random copy of a convex body intersects the integer lattice in a certain way. A conjecture by Erdos: We study the statement by Erdos "On every convex curve there exists a point P such that every circle with centre P intersects the curve in at most 2 points." A YaoYao type theorem: Given a nice measure in R^d, we show that there is a partition P of R^d into 3*2^(d/2) convex pieces of equal measure such that every hyperplane avoids at least 2 elements of P. Line transversals: Given a family F of balls in R^d such that every three have a transversal line, we bound the blowup factor l needed so that lF has a line transversal. Longest lattice convex chains: Given a triangle with two specified vertices v_1, v_2 in Z^2, we bound the size of the largest lattice convex chain from v_1 to v_2. The techniques used to tackle these problems are very diverse and include results from analysis, combinatorics, number theory and topology, as well as the use of computers.
Type:  Thesis (Doctoral) 

Title:  Problems in Convex Geometry 
Open access status:  An open access version is available from UCL Discovery 
Language:  English 
UCL classification:  UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences 
URI:  http://discovery.ucl.ac.uk/id/eprint/1388895 
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