In this paper we start to investigate a new body of questions in additive combinatorics. The
fundamental Cauchy--Davenport theorem gives a lower bound on the size of a sumset A+ B …

In this short survey we want to present some of the impact of Minkowski's successive minima
within Convex and Discrete Geometry. Originally related to the volume of an o-symmetric …

Let S be a discrete subset of R n and define c (S, k) as the smallest number with the property
that if a finite family of convex sets has exactly k points of S in common, then at most c (S, k) …

Gardner et al. posed the problem to find a discrete analogue of Meyer's inequality bounding
from below the volume of a convex body by the geometric mean of the volumes of its slices …

Let d≥ 3 be a natural number. We show that for all finite, non-empty sets that are not
contained in a translate of a hyperplane, we havewhere δ> 0 is an absolute constant only …

For a set A of points in the plane, not all collinear, we denote by tr (A) tr (A) the number of
triangles in a triangulation of A, that is, tr (A)= 2i+ b-2 tr (A)= 2 i+ b-2, where b and i are the …

We study upper bounds on the number of lattice points for convex bodies having their
centroid at the origin. For the family of simplices as well as in the planar case we obtain best …

Van der Corput's provides the sharp bound vol\nolimits (C) ≤ m 2^ d vol (C)≤ m 2 d on the
volume of ad-dimensional origin-symmetric convex body C that has 2 m-1 2 m-1 points of …

This thesis addresses classical lattice point problems in discrete and convex geometry.
Integer points in convex bodies are the central objects of our studies. In the second chapter …

In this thesis, we study a variety of problems at the interface of arithmetic combinatorics,
analytic number theory and harmonic analysis. We begin by considering the topic of …