Complex and quaternionic analogues of Busemann’s random simplex and intersection inequalities

Abstract In this paper, we extend two celebrated inequalities by Busemann—the random simplex inequality and the intersection inequality—to both complex and quaternionic vector spaces. Our proof leverages a monotonicity property under symmetrization with respect to complex or quaternionic hyperplanes. Notably, we demonstrate that the standard Steiner symmetrization, contrary to assertions in a paper by Grinberg, does not exhibit this monotonicity property.