On some determinant and matrix inequalities with a geometrical flavour

TL;DR

This paper explores determinant and matrix inequalities with geometric aspects, generalizing previous work, identifying optimizers, and deriving a new matrix inequality that extends existing results.

Contribution

It introduces a generalized framework for determinant inequalities, finds their optimizers, and presents a new geometric matrix inequality extending prior bounds.

Findings

Established optimizers for determinant inequalities.

Generalized inequalities related to Macbeath and Gressman.

Derived a new matrix inequality extending Christ's results.

Abstract

In this paper we study some determinant inequalities and matrix inequalities which have a geometrical flavour. We first examine some inequalities which place work of Macbeath [13] in a more general setting and also relate to recent work of Gressman [8]. In particular, we establish optimisers for these determinant inequalities. We then use these inequalities to establish our main theorem which gives a geometric inequality of matrix type which improves and extends some inequalities of Christ in [5].