TY  - UNPB
N1  - Thesis digitised by Proquest
EP  - 88
AV  - public
Y1  - 2005///
TI  - Combinatorial problems at the interface of discrete and convex geometry.
A1  - Prodromou, M.
M1  - Doctoral
UR  - https://discovery.ucl.ac.uk/id/eprint/1445801/
PB  - University of London
ID  - discovery1445801
N2  - 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.
ER  -