MPHA seminar, 12th October 2007, 1:50pm, Bloc627

Speaker: Boris Rubin, Louisiana State University
Title: Radon transforms and comparison of volumes.


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 known to be
 negative. For i=2 and i=3,  the problem is still open. However,  when
the body with smaller sections is a body of revolution,  the answer  is
affirmative. A complete solution to the problem will be presented in
the more general situation, when the body with smaller sections is invariant
under rotations, preserving mutually orthogonal subspaces of dimensions \ell
and n-\ell, respectively. The answer essentially depends on
\ell. The argument relies on the notion of canonical angles between
subspaces, spherical Radon transforms, and related harmonic analysis.