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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Intersection bodies and the Busemann-Petty problem


Author: R. J. Gardner
Journal: Trans. Amer. Math. Soc. 342 (1994), 435-445
MSC: Primary 52A38; Secondary 52A40
MathSciNet review: 1201126
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Abstract: It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in d-dimensional Euclidean space $ {\mathbb{E}^d}$ is negative for a given d if and only if certain centrally symmetric convex bodies exist in $ {\mathbb{E}^d}$ which are not intersection bodies. It is also shown that a cylinder in $ {\mathbb{E}^d}$ is an intersection body if and only if $ d \leq 4$, and that suitably smooth axis-convex bodies of revolution are intersection bodies when $ d \leq 4$. These results show that the Busemann-Petty problem has a negative answer for $ d \geq 5$ and a positive answer for $ d = 3$ and $ d = 4$ when the body with smaller sections is a body of revolution.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1201126-7
PII: S 0002-9947(1994)1201126-7
Keywords: Convex body, section, Busemann-Petty problem, intersection body
Article copyright: © Copyright 1994 American Mathematical Society