A generalization of the Busemann-Petty problem on sections of convex bodies
For a convex body \(K\) in Euclidean space \(\mathbb{R}^n\) and for a unit vector \(\xi\), the parallel section function \(A_{K,\xi}\) is defined by \[ A_{K,\xi}(z): =\text{vol}_{n-1} \bigl(K\cap (\xi^\perp+ z\xi)\bigr)\quad \text{for }z\in\mathbb{R}. \] The main result of this paper is the following theorem. Let \(K\) and \(L\) be \((k-1)\)-smooth origin symmetric convex bodies in \(\mathbb{R}^n\) such that, for every unit vector \(\xi\), \[ (-1)^{(k-1)/2}A_{k, \xi}^{(k-1)}(0)\leq(-1)^{(k-1)/2} A_{L,\xi}^{(k-1)}(0), \] where \(k\in\{1,\dots, n-1\}\) is an odd integer and \(f^{(k-1)}\) denotes the \((k-1)\)st derivative of \(f\). Then (i) if \(k\geq n-3\) then \(\text{vol}_n(K) \leq\text{vol}_n(L)\), (ii) if \(k<n-3\) then it is still possible that \(\text{vol}_n(K)> \text{vol}_n (L)\). This theorem is closely related to the (known) solution of the Busemann-Petty problem; in fact, it implies its affirmative answer for \(n=4\) and its negative answer for \(n=5\). The proof of the theorem uses Fourier analysis. A crucial fact in the solution of the Busemann-Petty problem was the connection between this problem and intersection bodies, due to Lutwak. The paper contains two further results generalizing this connection, as well as a generalization of intersection bodies.
scientific article; zbMATH DE number 1310633
scientific article