Generalizations of the Busemann-Petty problem for sections of convex bodies

Boris Rubin; Gaoyong Zhang

doi:10.1016/j.jfa.2003.10.008

Generalizations of the Busemann-Petty problem for sections of convex bodies

Boris Rubin, Gaoyong Zhang

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We present generalizations of the Busemann-Petty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For two- three-dimensional sections, both negative and positive answers are given depending on the orders of dual volumes involved, and certain cases remain open. For bodies of revolution, a complete solution is obtained in all dimensions.

Original language	English (US)
Pages (from-to)	473-501
Number of pages	29
Journal	Journal of Functional Analysis
Volume	213
Issue number	2
DOIs	https://doi.org/10.1016/j.jfa.2003.10.008
State	Published - Aug 15 2004

Keywords

Convex bodies
Dual volumes
The Busemann-Petty problem
The spherical Radon transform

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2003.10.008

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title = "Generalizations of the Busemann-Petty problem for sections of convex bodies",

abstract = "We present generalizations of the Busemann-Petty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For two- three-dimensional sections, both negative and positive answers are given depending on the orders of dual volumes involved, and certain cases remain open. For bodies of revolution, a complete solution is obtained in all dimensions.",

keywords = "Convex bodies, Dual volumes, The Busemann-Petty problem, The spherical Radon transform",

author = "Boris Rubin and Gaoyong Zhang",

note = "Funding Information: ·Corresponding author. E-mail addresses: boris@math.huji.ac.il (B. Rubin), gzhang@math.poly.edu (G. Zhang). 1Supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). 2Supported in part by NSF Grant DMS-0104363.",

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doi = "10.1016/j.jfa.2003.10.008",

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AU - Rubin, Boris

AU - Zhang, Gaoyong

N1 - Funding Information: ·Corresponding author. E-mail addresses: boris@math.huji.ac.il (B. Rubin), gzhang@math.poly.edu (G. Zhang). 1Supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). 2Supported in part by NSF Grant DMS-0104363.

PY - 2004/8/15

Y1 - 2004/8/15

N2 - We present generalizations of the Busemann-Petty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For two- three-dimensional sections, both negative and positive answers are given depending on the orders of dual volumes involved, and certain cases remain open. For bodies of revolution, a complete solution is obtained in all dimensions.

AB - We present generalizations of the Busemann-Petty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For two- three-dimensional sections, both negative and positive answers are given depending on the orders of dual volumes involved, and certain cases remain open. For bodies of revolution, a complete solution is obtained in all dimensions.

KW - Convex bodies

KW - Dual volumes

KW - The Busemann-Petty problem

KW - The spherical Radon transform

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JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

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ER -

Generalizations of the Busemann-Petty problem for sections of convex bodies

Abstract

Keywords

ASJC Scopus subject areas

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