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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 classical and spherical Radon transforms. In both cases we point out classes of functions for which the answer is affirmative, and show that in general the answer is negative if the functions do not belong to these classes. 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. We also deduce slicing inequalities that are related to the well-known Oberlin-Stein type estimates for the Radon transform.

An analogue of polynomially integrable bodies in even-dimensional spaces

https://doi.org/10.1016/j.jmaa.2023.127071

Agranovsky, M; Koldobsky, A; Ryabogin, D; Yaskin, V. (January 2024, Journal of mathematical analysis and applications)

A bounded domain $$K \subset R^n$$ is called polynomially integrable ifthe $(n-1)$-dimensional volume of the intersection $$K$$ with a hyperplane $$\Pi$$ polynomially depends on the distance from $$\Pi$$ to the origin. It was proved in \cite{KMY} that there are no such domains with smooth boundary if $$n$$ is even, and if $$n$$ is odd then the only polynomially integrable domains with smooth boundary are ellipsoids. In this article, we modify the notion of polynomial integrability for even $$n$$ and consider bodies for which the sectional volume function is a polynomial up to a factor which is the square root of a quadratic polynomial, or, equivalently, the Hilbert transform of this function is a polynomial. We prove that ellipsoids in even dimensions are the only convex infinitely smooth bodies satisfying this property.

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