Analysis Seminar

Thursday, October 19, 2023 - 3:30pm to 4:30pm

Michael Roysdon

Applied Mathematics and Statistics

The Busemann-Petty problem in Convex Geometry asks that if we have a pair of origin-symmetric convex bodies $K, L$ for which $\vol(K \cap H) \leq \vol_(L \cap H)$ for all (n-1)-dimensional subspaces $H$ in $\R^n$, does it necessarily follow that $\vol(K) \leq \vol(L)$? This question originally asked in 1956 wasn't resolved until the end of the 1990s; its solution involves a very special class of convex bodies called intersection bodies introduced by Lutwak in 1988 as well as the connection between Harmonic Analysis and Convex Geometry discovered at the end of the 1990s. The answer to the Busemann-Petty Problem is affirmative when $n \leq 4$ and negative whenever $n \geq 5$. Inspired by this problem and its solution, we examine the following question: given a pair of non-negative, even and continuous functions $f$ and $g$ such that the Radon transform (spherical Radon transform) of f is pointwise smaller than the Radon transform (spherical Radon transform) of g, does it follow that the L^p-norm of f is smaller than the L^p norm of g whenever p>0? If time permits, we will discuss the isomorphic Busemann-Petty problem, which is equivalent to Bourgain's slicing problem, and L^p-L^q-estimates of reverse Oberlin-Stein type for the spherical Radon transform. This is based on the a joint work with Alexander Koldobsky and Artem Zvavitch.