Inequalities for the Radon Transform on Convex Sets

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