| The thesis is devoted to the study of various problems arising from Convex Geometry and Geometric Functional Analysis using tools of Fourier Analysis.;In chapters two through four we consider the Busemann-Petty problem and its different modifications and generalizations. We solve the Busemann-Petty problem in hyperbolic and spherical spaces, and the lower dimensional Busemann-Petty problem in the hyperbolic space. In the Euclidean space we modify the assumptions of the original Busemann-Petty problem to guarantee the affirmative answer in all dimensions.;In chapter five we introduce the notion of embedding of a normed space in L0, investigate the geometry of such spaces and prove results confirming the place of L0 in the scale of Lp spaces.;Chapter six is concerned with the study Lp-centroid bodies associated to symmetric convex bodies and generalization of some known results of Lutwak and Grinberg, Zhang to the case -1 < p < 1.;In chapter seven we discuss Khinchin type inequalities and the slicing problem. We obtain a version of such inequalities for p > -2 and as a consequence we prove the slicing problem for the unit balls of spaces that embed in Lp, p > -2. |
