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Combinatorial problems at the interface of discrete and convex geometry.

Prodromou, M.; (2005) Combinatorial problems at the interface of discrete and convex geometry. Doctoral thesis , University of London. Green open access

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Abstract

This thesis consists of three chapters. The first two chapters concern lattice points and convex sets. In the first chapter we consider convex lattice polygons with minimal perimeter. Let n be a positive integer and any norm in R2. Denote by B the unit ball of and Vb,u the class of convex lattice polygons with n vertices and least -perimeter. We prove that after suitable normalisation, all members of Vb,u tend to a fixed convex body, as n > oo. In the second chapter we consider maximal convex lattice polygons inscribed in plane convex sets. Given a convex compact set K CM2 what is the largest n such that K contains a convex lattice n-gon We answer this question asymptotically. It turns out that the maximal n is related to the largest affine perimeter that a convex set contained in K can have. This, in turn, gives a new characterisation of Ko, the convex set in K having maximal affine perimeter. In the third chapter we study a combinatorial property of arbitrary finite subsets of Rd. Let X C Rd be a finite set, coloured with J colours. Then X contains a rainbow subset 7 CX, such that any ball that contains Y contains a positive fraction of the points of X.

Type: Thesis (Doctoral)
Title: Combinatorial problems at the interface of discrete and convex geometry.
Identifier: PQ ETD:593125
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Thesis digitised by Proquest
UCL classification: UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: http://discovery.ucl.ac.uk/id/eprint/1445801
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