Convex body, Section, Busemann-Petty problem, Intersection body
It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in d-dimensional Euclidean space Ed is negative for a given d if and only if certain centrally symmetric convex bodies exist in Ed which are not intersection bodies. It is also shown that a cylinder in Ed is an intersection body if and only if d ≤ 4, and that suitably smooth axis-convex bodies of revolution are intersection bodies when d ≤ 4. These results show that the Busemann-Petty problem has a negative answer for d ≥ 5 and a positive answer for d = 3 and d = 4 when the body with smaller sections is a body of revolution.
Transactions of the American Mathematical Society
Required Publisher's Statement
First published in Transactions of the American Mathematical Society in Volume 342, Number 1, March 1994, published by the American Mathematical Society
Gardner, Richard J., "Intersection Bodies and the Busemann-Petty Problem" (1994). Mathematics. 24.