Applications of fourier analysis to intersection bodies

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 k-intersection bodies is related to the generalization of the Busemann-Petty problem. We are interested in comparing classes of k-intersection bodies. In the first chapter we present the result that was published in J. Schlieper, A note on k-intersection bodies, Proc. Amer. Math. Soc., 135 (2007), 2081-2088. The result examines the conjecture that the classes of k-intersection bodies increase with k. In particular, the result constructs a 4intersection 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 ‖x‖π(a) = (ai1x q 1 + · · · ainxn) where a = (a1, . . . , an) with a1 ≥ · · · ≥ an > 0 and π(a) is a permutation of the vector a . Define a 1-homogeneous functional based on this family of norms as follows: for k ≥ 1, ‖x‖k = (∑