This is an announcement for the paper "Generalized intersection bodies" by Emanuel Milman.

Abstract: We study the structures of two types of generalizations of intersection-bodies and the problem of whether they are in fact equivalent. Intersection-bodies were introduced by Lutwak and played a key role in the solution of the Busemann-Petty problem. A natural geometric generalization of this problem considered by Zhang, led him to introduce one type of generalized intersection-bodies. A second type was introduced by Koldobsky, who studied a different analytic generalization of this problem. Koldobsky also studied the connection between these two types of bodies, and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized Busemann-Petty problem. We show that these classes share many identical structure properties, proving the same results using Integral Geometry techniques for Zhang's class and Fourier transform techniques for Koldobsky's class. Using a Functional Analytic approach, we give several surprising equivalent formulations for the equivalence problem, which reveal a deep connection to several fundamental problems in the Integral Geometry of the Grassmann Manifold.

Archive classification: Functional Analysis; Geometric Topology; Metric Geometry

Remarks: 44 pages

The source file(s), generalized-intersection-bodies-for-arxiv.bbl: 6010 bytes, generalized-intersection-bodies-for-arxiv.tex: 129282 bytes, is(are) stored in gzipped form as 0512058.tar.gz with size 38kb. The corresponding postcript file has gzipped size 159kb.

Submitted from: emanuel.milman@weizmann.ac.il

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