Intersection Bodies and Generalized Cosine Transforms

Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. We review some known facts and give them new proofs. The main focus is interrelation between generalized cosine transforms of different kinds and their application to investigation of certain family of intersection bodies, which we call λ-intersection bodies. The latter include k-intersection bodies (in the sense of A. Koldobsky) and unit balls of finite-dimensional subspaces of Lp-spaces. In particular, we show that restriction of the spherical Radon transforms and the generalized cosine transforms onto lower dimensional subspaces preserves their integralgeometric structure. We apply this result to the study of sections of λ-intersection bodies. A number of new characterizations of this class of bodies and examples are given.