Intersection bodies and the Busemann-Petty problem

Gardner, R. J.

doi:10.1090/S0002-9947-1994-1201126-7

Intersection bodies and the Busemann-Petty problem
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by R. J. Gardner

Trans. Amer. Math. Soc. 342 (1994), 435-445

DOI: https://doi.org/10.1090/S0002-9947-1994-1201126-7

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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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Intersection bodies and the Busemann-Petty inequalities in