Applications of the fourier transform to convex geometry

Abstract

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 L[0], investigate the geometry of such spaces and prove results confirming the place of L[0] in the scale of L [subscript p] spaces. Chapter six is concerned with the study L [subscript p]-centroid bodies associated to symmetric convex bodies and generalization of some known results of Lutwak and Grinberg, Zhang to the case [minus] 1 [less than] p [less than] 1. In chapter seven we discuss Khinchin type inequalities and the slicing problem. We obtain a version of such inequalities for p [greater than] [minus] 2 and as a consequence we prove the slicing problem for the unit balls of spaces that embed in L[subscript] p, p [greater than] [minus] 2.