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A comment on the low-dimensional Busemann-Petty problem

Emanuel Milman
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Abstract

The generalized Busemann-Petty problem asks whether centrally-symmetric convex bodies having larger volume of all m-dimensional sections necessarily have larger volume. When m>3 this is known to be false, but the cases m=2,3 are still open. In those cases, it is shown that when the smaller body's radial function is a (n-m)-th root of the radial function of a convex body, the answer to the generalized Busemann-Petty problem is positive (for any larger star-body). Several immediate corollaries of this observation are also discussed.

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Emanuel Milman. 2005-12-10. A comment on the low-dimensional Busemann-Petty problem. https://arxiv.org/abs/math/0512208

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