Inequalities for the Radon Transform on Convex Sets
Abstract We prove an inequality that unifies previous works of the authors on the properties of the Radon transform on convex bodies including an extension of the Busemann–Petty problem and a slicing inequality for arbitrary functions. Let $K$ and $L$ be star bodies in ${\mathbb R}^n,$ let $0<k<n$ be an integer, and let $f,g$ be non-negative continuous functions on $K$ and $L$, respectively, so that $\|g\|_\infty =g(0)=1.$ Then \begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{n-k}n}|K|^{\frac kn}} \le \frac n{n-k} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}where $|K|$ stands for volume of proper dimension, $C$ is an absolute constant, the maximum is taken over all $(n-k)$-dimensional subspaces of ${\mathbb R}^n,$ and $d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)$ is the outer volume ratio distance from $K$ to the class of generalized $k$-intersection bodies in ${\mathbb R}^n.$ Another consequence of this result is a mean value inequality for the Radon transform. We also obtain a generalization of the isomorphic version of the Shephard problem.
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NSF-PAR ID:
10320829
Journal Name:
International Mathematics Research Notices
ISSN:
1073-7928
National Science Foundation
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2. Abstract

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3. Abstract

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5. (Ed.)
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