Generalized Intersection Bodies are not Equivalent

Details Generalized Intersection Bodies are not Equivalent Generalized Intersection Bodies are not Equivalent

2007-01-26

arxiv.org/abs/math/0701779v2

Mathematics

Functional Analysis

18 pages, added a section with equivalent formulations using Fourier Transforms and Embeddings into L_p for p<0

Scientific paper

In 2000, A. Koldobsky asked whether two types of generalizations of the notion of an intersection-body, are in fact equivalent. The structures of these two types of generalized intersection-bodies have been studied by the author in [http://www.arxiv.org/math.MG/0512058], providing substantial positive evidence for a positive answer to this question. The purpose of this note is to construct a counter-example, which provides a surprising negative answer to this question in a strong sense. This implies the existence of non-trivial non-negative functions in the range of the spherical Radon transform, and the existence of non-trivial spaces which embed in L_p for certain negative values of p.

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