Remarks about Mixed Discriminants and Volumes

TL;DR

This paper establishes new inequalities for mixed discriminants and volumes of convex sets, linking them to an information functional analogous to Fisher information, enhancing understanding in convex geometry and matrix analysis.

Contribution

It introduces novel inequalities for mixed discriminants and volumes, and explores their connection to a geometric analogue of Fisher information.

Findings

Proved inequalities for mixed discriminants of positive semi-definite matrices.

Established inequalities for mixed volumes of convex sets.

Connected mixed volumes to a monotonicity property of an information functional.

Abstract

In this note we prove certain inequalities for mixed discriminants of positive semi-definite matrices, and mixed volumes of compact convex sets in n-dimensions. Moreover, we discuss how the latter are related to the monotonicity of an information functional on the class of convex bodies, which is a geometric analogue of the classical Fisher information.