The Busemann--Petty problem in hyperbolic and spherical spaces (Q2497330)

Let \(K\) and \(L\) be two convex origin-symmetric bodies in \(R^n\). Let \(H\) be an arbitrary hyperplane in \(R^n\) passing through the origin. Suppose that \[ \text{ vol}_{n-1} (K \cap H) \leq \text{ vol}_{n-1} (L \cap H) \tag{*} \] for any \(H\) where \(\text{ vol}_{n-1}\) means the \((n-1)\)-dimensional volume. Is then \(\text{ vol}_n(K) \leq \text{ vol}_n(L)\)? This is the Busemann-Petty problem. The answer to this question is known to be positive for \(n\leq 4\) and negative for \(n \geq 5\). The author considers this problem in spherical space \(S^n\) and hyperbolic space \(H^n\). He shows that the answer in \(S^n\) is the same as in \(R^n\): positive for \(n\leq 4\) and negative for \(n \geq 5\). He establishes however that the things are different in \(H^n\). For \(n=2\), the answer is still positive since condition (*) implies that \(K \subset L\). But for \(n \geq 3\), the following result is proved. Theorem. There are convex centrally symmetric bodies \(K,L \subset H^n\), \(n \geq 3\), such that the condition (*) holds for every central totally-geodesic hyperplane \(H\) in \(H^n\), but \(\text{ vol}_n(K) > \text{ vol}_n(L).\)