Inequalities for the derivatives of the Radon transform on convex bodies

TL;DR

This paper extends known bounds on the Radon transform of probability densities to its derivatives on convex bodies and introduces a comparison theorem for these derivatives, advancing understanding of geometric analysis.

Contribution

It provides new lower bounds for the derivatives of the Radon transform and establishes a comparison theorem, broadening the scope of Radon transform inequalities.

Findings

Lower bounds for derivatives of the Radon transform on convex bodies

A comparison theorem for these derivatives

Extension of previous bounds to derivatives

Abstract

It has been proved that the sup-norm of the Radon transform of an arbitrary probability density on an origin-symmetric convex body of volume 1 is bounded from below by a positive constant depending only on the dimension. In this note we extend this result to the derivatives of the Radon transform. We also prove a comparison theorem for these derivatives.