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^*

^*Corresponding author for this work

Einstein Institute of Mathematics

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

44 Scopus citations

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
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 - 15 Aug 2004

Keywords

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

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10.1016/j.jfa.2003.10.008

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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.",

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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.

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KW - Dual volumes

KW - The Busemann-Petty problem

KW - The spherical Radon transform

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

Abstract

Keywords

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