An inequality between the parts into which a convex body is divided by a plane section
\textit{J. Bokowski} and \textit{E. Sperner jun.} [J. Reine Angew. Math. 311/312, 80-100 (1979; Zbl 0415.52008) found an upper bound for the product of the volumes of the two parts into which a hyperplane \(L_ n- 1\) splits a convex body K in Euclidean n-space. The bound is a constant times \(D^{n+1}\sigma_ n-1\) where D is the diameter of K, \(\sigma_ n-1\) is the (n-1)-dimensional volume of \(K\cap L_ n-1\). The present paper provides a different, integral-geometric proof for the cases \(n=2, 3\) and this proof is generalized to provide the corresponding inequalities in the cases of elliptic and hyperbolic spaces of dimensions 2 and 3.
scientific article; zbMATH DE number 3842626