Comparison of volumes of convex bodies in real, complex, and quaternionic spaces (Q2637919)

The (meanwhile solved) Busemann-Petty problem asked whether two origin-symmetric convex bodies \(K,L\) in \({\mathbb R}^n\) satisfying \(\mathrm{vol}_{n-1}(K\cap H)\leq \mathrm{vol}_{n-1}(L\cap H)\) for any hyperplane \(H\) through the origin also satisfy \(\mathrm{vol}_{n}(K)\leq \mathrm{vol}_{n}(L)\). This paper presents a thorough unified treatment of analogs and extensions of the Busemann-Petty problem in real, complex and quaternionic \(n\)-dimensional spaces. The bodies under consideration are now `equilibrated' (absolutely convex in the complex case). In the quaternionic case, it must be distinguished between left and right vector spaces. Volumes are defined via the natural bijection \({\mathbb K}^n\to{\mathbb R}^N\), where \({\mathbb K}^n \in \{{\mathbb R}^n, {\mathbb C}^n, {\mathbb H}^n_{\mathrm{left}}, {\mathbb H}^n_{\mathrm{right}}\}\) and \(N= dn\) with \(d=1,2,4\), corresponding to the real, complex, and quaternionic case, respectively. The different cases are treated simultaneously; the method of proof (other than in the previous publications on the topic) relies on properties of the generalized cosine transforms on the unit sphere. A connection is made with certain lower dimensional slice comparison problems for \(G\)-invariant convex bodies in \({\mathbb R}^N\), \(N= dn\), where \(G\) is a suitable subgroup of \(SO(N)\). The main result says that the hyperplane slice comparison problem in \({\mathbb K}^n\) has an affirmative answer if and only if \(n\leq 2+2/d\), with \(d\) as above. Further, generalized lower dimensional section comparison problems are treated, where comparison of slices is realized in terms of certain derivatives of slicing functions.