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.