The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry (Q2655759)

The lower dimensional Buseman-Petty problem asks whether \(n\)-dimensional centrally symmetric convex bodies with smaller \(i\)-dimensional central sections necessarily have smaller volumes. The answer to this problem is known to be positive for \(i=1\), and negative for \(i>3\), but it is still open for \(i=2,3\), with \(n>4\). The answer is also known to be positive when the body with smaller sections is a body of revolution. This paper extends this result to the case where the body with smaller sections is invariant under rotations, preserving mutually orthogonal subspaces of dimensions \(l\) and \(n-l\), so that \(i+l\leq n\). The author considers the subgroup of orthogonal transformations \[ K_l=\left\{ \gamma \in O(n):\gamma=\left[ \begin{matrix} \alpha & 0 \\ 0 & \beta \end{matrix} \right],\;\alpha\in O(n-l),\;\beta\in O(l) \right\}. \] A star body \(A\) is \(K_l\)-symmetric if \(\gamma A= A\) for all \(\gamma \in K_l\). Clearly, in this case, \(A=-A\). Let \(x=(x',x'')\in \mathbb R^n\), with \(x'\in \mathbb R^{n-l}\), \(x''\in \mathbb R^l\). The \((q,l)\)-ball \(B_{q,l}^n=\{x: | x'| ^q+ | x'' | ^q \leq 1\}\), with \(q>0\), is an example of a \(K_l\)-symmetric star body. The basic idea of this new approach is observing that the relative position of two subspaces is determined by a finite number of canonical angles \(\omega_1,\dots,\omega_m\). Let \(G_{n,i}\) be the Grassmann manifold of the \(i\)-dimensional linear subspaces of \(\mathbb R^n\), \(\text{vol}_i(.)\) denote the \(i\)-dimensional volume function, and \(G_{n,i}^l=\{ \xi \in G_{n,i}: \omega_1=\cdots=\omega_m \}\). The main result is the following: Let \(1\leq l \leq n/2\), \(i+l\leq n\), and let \(A\) be a \(K_l\)-symmetric star body in \(\mathbb R^n\). {\parindent6mm \begin{itemize}\item[a)] If \(1\leq i\leq l\), then the implication \[ \text{vol}_i(A\cap \xi) \leq\text{vol}_i(B\cap \xi), \xi \in G_{n,i}^l \Rightarrow \text{vol}_n(A) \leq\text{vol}_n(B) \] is true for every origin symmetric star body \(B\). \item[b)] If \(i=l+1\) or \(i=l+2\), then the above implication holds for every origin symmetric star body \(B\) provided that \(A\) is convex. \end{itemize}} The author also proves the following negative result: If \(i>l+2\), and \(B=B_{4,l}^n\), then there is an infinitely smooth \(K_l\)-symmetric convex body \(A\), such that \(\text{vol}_i(A\cap \xi)\leq\text{vol}_i(B\cap \xi)\) for all \(\xi \in G_{n,i}\), but \(\text{vol}_n(A)>\text{vol}_n(B)\). The arguments involved in the proofs use spherical Radon transforms, properties of intersection bodies and the generalized cosine transforms.