Analysis seminar
Speaker: Alex Koldobsky (University of Missouri Columbia)
Title: Comparison problems for the Radon transform
Abstract: Given two non-negative functions $f$ and $g$ such that the Radon transform of $f$ is pointwise smaller than the Radon transform of $g$, does it follow that the $L^p$-norm of $f$ is smaller than the $L^p$-norm of $g$ for a given $p>1?$ We consider this problem for the spherical and spatial Radon transforms. In both cases we point out classes of functions for which the answer is affirmative. The results are in the spirit of the solution of the Busemann-Petty problem from convex geometry, and the classes of functions that we introduce generalize the class of intersection bodies introduced by Lutwak in 1988.