Generalized intersection bodies

By Emanuel Milman. 2005

66 citations recorded by the source corpus.

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 structural 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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5a2de39718fa3e00a9040af2e4b5e33229c1c9b2

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In Applied Mathematics

In Astronomy and Astrophysics

Emanuel Milman

In Geometry and Topology

Citations in the corpus

Dual mixed volumes and the slicing problem (2005, background): “[Mil05] for an account of recent progress in this direction).”
Adams inequalities on measure spaces (2009, unknown): “<n), then Z Sn−1 E∧ −d(f)E∧ d−n(g) = Z Sn−1 fg. (86) The formula was originally found by Koldobsky in case fand gare powers of Minkowsky functionals of smooth star bodies (see [K] for a proof). In [Mil] the above version is proven for f,greal-valued and even, but a small modiﬁcation of the proof ”