 Award ID(s):
 2001128
 Publication Date:
 NSFPAR ID:
 10225264
 Journal Name:
 Forum of Mathematics, Sigma
 Volume:
 9
 ISSN:
 20505094
 Sponsoring Org:
 National Science Foundation
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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 nonnegative continuous functions on $K$ and $L$, respectively, so that $\g\_\infty =g(0)=1.$ Then $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{nk}n}K^{\frac kn}} \le \frac n{nk} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}$$where $K$ stands for volume of proper dimension, $C$ is an absolute constant, the maximum is taken over all $(nk)$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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Abstract Let $S$ be a scheme and let $\pi : \mathcal{G} \to S$ be a ${\mathbb{G}}_{m,S}$gerbe corresponding to a torsion class $[\mathcal{G}]$ in the cohomological Brauer group ${\operatorname{Br}}^{\prime}(S)$ of $S$. We show that the cohomological Brauer group ${\operatorname{Br}}^{\prime}(\mathcal{G})$ of $\mathcal{G}$ is isomorphic to the quotient of ${\operatorname{Br}}^{\prime}(S)$ by the subgroup generated by the class $[\mathcal{G}]$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes.

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