Complex and Quaternionic Analogues of Busemann's Random Simplex and Intersection Inequalities

TL;DR

This paper extends Busemann's inequalities to complex and quaternionic vector spaces, revealing new symmetrization properties and correcting misconceptions about Steiner symmetrization.

Contribution

It introduces complex and quaternionic analogues of Busemann's inequalities and clarifies the monotonicity properties of symmetrization methods.

Findings

Extended Busemann's inequalities to complex and quaternionic spaces

Disproved Steiner symmetrization's monotonicity in these contexts

Provided new insights into symmetrization properties

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.