Intersection bodies and generalized cosine transforms

摘要

The paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis. A new concept of lambda-intersection body, that unifies some known classes of geometric objects, is introduced. A parallel between trace theorems in function theory, restriction onto lower-dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms, and sections of lambda-intersection bodies is established. New integral formulas for different classes of cosine transforms are obtained and examples of lambda-intersection bodies are given. We also revisit some known facts in this area and give them new simple proofs. (c) 2008 Elsevier Inc. All rights reserved.