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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The research was supported in part by the NSF grant DMS-0556157 and the Louisiana EPSCoR program, sponsored by NSF and the Board of Regents Support Fund.
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Rubin, B. The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry. Isr. J. Math. 173, 213–233 (2009). https://doi.org/10.1007/s11856-009-0089-7
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DOI: https://doi.org/10.1007/s11856-009-0089-7