Generalized intersection bodies (Q860791)

In connection with the generalized Busemann-Petty problem in \({\mathbb R}^n\) for sections of codimension \(k\in \{1,\dots,n-1\}\), two classes of generalized intersection bodies have been introduced. A body \(K\) (i.e., a \(0\)-symmetric star body with continuous radial function \(\rho_K\)) belongs to the class \({\mathcal BP}^n_k\) (introduced by \textit{G. Zhang} [Am. J. Math. 118, 319--340 (1996; Zbl 0854.52004)]) if \(\rho_K^k\) is a (suitably generalized) Radon transform of a non-negative Borel measure on the Grassmannian \(G(n,n-k)\). A star body \(K\) is a \(k\)-intersection body of a star body \(L\) if \(\text{ Vol}(K\cap H^\bot)=\text{ Vol}(L\cap H)\) for every \(H\in G(n,n-k)\). The body \(K\) belongs to the class \({\mathcal I}^n_k\) (introduced by \textit{A. Koldobsky} [Geom. Funct. Anal. 10, No. 6, 1507--1526 (2000; Zbl 0974.52002)]) if it is a limit, in the radial metric, of \(k\)-intersection bodies of star bodies. It is known that \({\mathcal BP}^n_k\subset {\mathcal I}^n_k\) and that a proof of \({\mathcal BP}^n_k = {\mathcal I}^n_k\) would settle the remaining open cases of the generalized Busemann-Petty problem. The present paper first supports the conjectured equality \({\mathcal BP}^n_k = {\mathcal I}^n_k\) by establishing the same structural properties for the classes \({\mathcal BP}^n_k\) and \({\mathcal I}^n_k\). The proofs use different tools, integral geometry of Grassmannians in the first case and Fourier transforms of homogeneous distributions in the second case. After discussing some consequences, the paper proceeds with using the previous results and a functional analytic approach to establish conditions in terms of properties of generalized Radon transforms that are equivalent to \({\mathcal BP}^n_k = {\mathcal I}^n_k\).