Generalizations of the Busemann-Petty problem for sections of convex bodies (Q1883600)

The authors formulate a problem on dual volumes and the following generalized version of it. Let \(K\) and \(L\) be origin-symmetric convex bodies in \(\mathbb{R}^n\), let \(0< k< l<\infty\), and define \[ I_k(K,\xi)= \int_{S^{n-1}\cap\xi} \rho_K(u)^k \,d_\xi u,\quad J_l(K)= \int_{S^{n-1}} \rho_K(u)^l \,du. \] Here \(\rho_K\) is the radial function of \(K\) on the unit sphere \(S^{n-1}\), and \(\xi\in G_{n,i}\), the Grassmannian of \(i\)-dimensional linear subspaces of \(\mathbb{R}^n\). Problem B: If \(I_k(K,\xi)\leq I_k(L,\xi)\) for all \(\xi\in G_{n,i}\) and some integer \(i\) with \(2\leq i\leq n-1\), does it follow that \(J_l(K)\leq J_l(L)\)? For \(k= i= n- 1\), \(l= n\), this is the Busemann-Petty problem. The known answer to this problem is considerably extended by the results of this paper: (a) Problem B has a negative answer in the cases (i) \(i\geq 4\); (ii) \(l- k> n- i\) for \(i= 2\) or \(3\). (b) Problem B has a positive answer when \(l= k+ 1\) for \(i= 2\) or \(3\). (c) Other cases of \(i= 2\) or \(3\) remain open in general, but have positive answers when \(K\) is a body of revolution. Main tools of the proofs are totally geodesic Radon transforms and harmonic analysis on the sphere.