The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry |
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Authors: | Boris Rubin |
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Institution: | 1.Department of Mathematics,Louisiana State University,Baton Rouge,USA |
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Abstract: | The lower dimensional Busemann-Petty problem asks, whether n-dimensional centrally symmetric convex bodies with smaller i-dimensional central sections necessarily have smaller volumes. For i = 1, the affirmative answer is obvious. If i > 3, the answer is negative. For i = 2 or i = 3 (n > 4), the problem is still open, however, when the body with smaller sections is a body of revolution, the answer is affirmative.
The paper contains a solution to the problem in the more general situation, when the body with smaller sections is invariant
under rotations, preserving mutually orthogonal subspaces of dimensions ℓ and n − ℓ, respectively, so that i + ℓ ≤ n. The answer essentially depends on ℓ. The argument relies on the notion of canonical angles between subspaces, spherical Radon transforms, properties of intersection
bodies, and the generalized cosine transforms. |
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