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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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Keywords: Convex body, section, Busemann-Petty problem, intersection body
Article copyright: © Copyright 1994 American Mathematical Society