Topics in Convex Geometric Analysis and Discrete Tomography

Abstract

In this thesis, some topics in convex geometric analysis and discrete tomography are studied. Firstly, let K be a convex body in the n-dimensional Euclidean space. Is K uniquely determined by its sections? There are classical results that explain what happens in the case of sections passing through the origin. However, much less is known about sections that do not contain the origin. Here, several problems of this type and the corresponding uniqueness results are established. We also establish a discrete analogue of the Aleksandrov theorem for the areas and the surface areas of projections. Finally, we find the best constant for the Grünbaum’s inequality for projections, which generalizes both Grünbaum’s inequality, and an old inequality of Minkowski and Radon.