[Banach] Abstract of a paper by Emanuel Milman
alspach at www.math.okstate.edu
Mon Dec 5 07:25:26 CST 2005
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;
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 at weizmann.ac.il
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