Applications of fourier analysis to intersection bodies

Abstract

The concept of an intersection body is central for the dual Brunn-Minkowski theory and has played an important role in the solution of the Busemann-Petty problem. A more general concept of [kappa]-intersection bodies is related to the generalization of the Busemann-Petty problem. We are interested in comparing classes of [kappa]-intersection bodies. In the first chapter we present the result that was published in J. Schlieper, A note on [kappa]-intersection bodies, Proceedings American. Mathematical Society,135 (2007), 2081-2088. The result examines the conjecture that the classes of [kappa]-intersection bodies increase with [kappa]. In particular, the result constructs a 4- intersection body that is not a 2-intersection body. The second chapter is concerned with the geometry of spaces of Lorentz type. We define a 1-homogeneous functional based on Lorentz type norms. Consider the family of norms [nearest integer function]x[nearest integer function][pi][alpha] = [alpha]i₁xq₁+ [alpha]inxqn₁/q where [alpha] = ([alpha]₁, . . . , [alpha]n) with [alpha]₁ [pi], . . . ,[pi] [alpha] [less than]0 and [pi]([alpha]) is [alpha] permutation of the vector [alpha]. Define a 1-homogeneous functional based on this family of norms as follows ... We examine the geometric properties of the space (Rn, [kappa].[kappa]k). First, we determine the conditions when the star body (Rn, [kappa].[kappa]) is a [kappa]-intersection body. Second, we find the extremal sections of the star body (Rn, [kappa].[kappa]). Throughout this work we use the Fourier Analytic methods that were recently developed.