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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凸集上 Radon 变换的不等式

我们证明了一个不等式，它统一了作者以前关于凸体上 Radon 变换的性质的工作，包括 Busemann-Petty 问题的扩展和任意函数的切片不等式。令$K$ 和$L$ 为${\mathbb R}^n,$ 中的星体 令$0<k<n$ 为整数，令$f,g$ 为$K 上的非负连续函数$ 和 $L$，使得 $\|g\|_\infty =g(0)=1.$ 然后 $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{nk}n}|K|^{\frac kn}} \le \frac n{nk} \left(d_{\textrm{ovr}}(K,\mathcal{B} \mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*} $$其中$|K|$代表适当维度的体积，$C$是一个绝对常数，最大值取自${\mathbb R}^n,$和$的所有$(nk)$维子空间d_{\textrm{ovr}}(K, \mathcal{B}\mathcal{P}_k^n)$ 是从 $K$ 到 ${\mathbb R}^n 中广义 $k$-相交体类的外体积比距离。这个结果是 Radon 变换的平均值不等式。我们还获得了 Shephard 问题的同构版本的一般化。