Intersection Bodies and Generalized Cosine Transforms

TL;DR

This paper explores the relationships between generalized cosine transforms and intersection bodies, introducing the class of $\\lambda$-intersection bodies and providing new characterizations and proofs related to their geometric properties.

Contribution

It establishes the interrelation of different generalized cosine transforms and applies these to characterize and analyze $\\lambda$-intersection bodies, including new proofs and examples.

Findings

Restrictions of Radon and cosine transforms preserve geometric structure.

New characterizations of $\\lambda$-intersection bodies are provided.

Examples illustrate the properties of these bodies.

Abstract

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. The main focus of this article is interrelation between generalized cosine transforms of different kinds in the context of their application to investigation of a certain family of intersection bodies, which we call $\lam$-intersection bodies. The latter include $k$-intersection bodies (in the sense of A. Koldobsky) and unit balls of finite-dimensional subspaces of $L_p$-spaces. In particular, we show that restrictions onto lower dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms preserve their integral-geometric structure. We apply this result to the study of sections of $\lam$-intersection bodies. New characterizations of this class of bodies are obtained and examples are given. We also review some known facts and give them new proofs.