Alexander Koldobsky

Generalizing the slicing inequality for functions on convex bodies from [11], it was proved in [4] that there exists an absolute constant $c$ so that for any $n\in \mathbb N$, any $q\in [0,n-1)$ which is not an odd integer, any origin-symmetric convex body $K$ of volume one in $\mathbb R^n$ and any infinitely smooth probability density $f$ on $K$ we have $$\max_{ξ\in S^{n-1}} {\frac 1{\cos(πq/2)}\mathcal R f(ξ, \cdot)_t^{(q)}(0)} \ge \left( \frac {c(q+1)}{n}\right)^{\frac{q+1}2}.$$ Here $\mathcal R f(ξ,t)$ is the Radon transform of $f$, and the fractional derivative of the order $q$ is taken with respect to the variable $t\in \mathbb R$ with fixed $ξ\in S^{n-1}.$ In this note we show that there exist an origin-symmetric convex body $K$ of volume 1 in $\mathbb R^n$ and a continuous probability density $g$ on $K$ so that $$\max_{ξ\in S^{n-1}} {\frac 1{\cos(πq/2)}\mathcal R g(ξ, \cdot)_t^{(q)}(0)} \leq \frac 1{\sqrt n} (c(q+1))^{\frac{q+1}2}.$$ In the case $q=0$ this was proved in [5,6], and it was used there to obtain a lower estimate for the maximal outer volume ratio distance from an arbitrary origin-symmetric convex body $K$ to the class of intersection bodies. We extend the latter result to the class $L_{-1-q}^n$ of bodies in $\mathbb R^n$ that embed in $L_{-1-q}.$ Namely, for every $q\in [0,n)$ there exists an origin-symmetric convex body $K$ in $\mathbb R^n$ so that ${d_{\operatorname{ovr}}}(K, L_{-1-q}^n) \ge c n^{\frac 1{2(q+1)}}.$

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 $p>0$? 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.

A bounded domain $K \subset \mathbb R^n$ is called polynomially integrable if the $(n-1)$-dimensional volume of the intersection $K$ with a hyperplane $\Pi$ polynomially depends on the distance from $\Pi$ to the origin. It was proved in [7] 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.

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.

We prove new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing and distance inequalities. We also prove a sharp upper estimate for the outer volume ratio distance from an arbitrary convex body to the unit balls of subspaces of $L_p$.

For $p\ge 1$ , $n\in \mathbb {N}$ , and an origin-symmetric convex body $K$ in $\mathbb {R}^n,$ let $\begin{equation*} d_\textrm {{ovr}}(K,L_p^n) = \inf \left \{ \Big (\frac {|D|}{|K|}\Big )^{1/n}: K \subseteq D,\ D\in L_p^n \right \} \end{equation*}$ be the outer volume ratio distance from $K$ to the class $L_p^n$ of the unit balls of $n$ -dimensional subspaces of $L_p.$ We prove that there exists an absolute constant $c>0$ such that $\begin{equation*}\frac {c\sqrt {n}}{\sqrt {p\log \log n}}\le \sup _K d_\textrm {{ovr}}(K,L_p^n)\le \sqrt {n}. \end{equation*}$ This result follows from a new slicing inequality for arbitrary measures, in the spirit of the slicing problem of Bourgain. Namely, there exists an absolute constant $C>0$ so that for any $p\ge 1,$ any $n\in \mathbb {N}$ , any compact set $K \subseteq \mathbb {R}^n$ of positive volume, and any Borel measurable function $f\ge 0$ on $K$ ,