# Pfaffians, hafnians and products of real linear functionals

## TL;DR

This paper extends classical inequalities to pfaffians and hafnians, and applies these results to improve bounds on the norms of products of linear functionals in real Euclidean spaces.

## Contribution

It introduces pfaffian and hafnian versions of Lieb's inequalities and uses them to enhance bounds in the real linear polarization constant problem.

## Key findings

- Established pfaffian and hafnian inequalities for positive semi-definite matrices.
- Improved the lower bound on the norm of products of linear functionals.
- Connected pfaffian/hafnian inequalities to the real linear polarization constant problem.

## Abstract

We prove pfaffian and hafnian versions of Lieb's inequalities on determinants and permanents of positive semi-definite matrices. We use the hafnian inequality to improve the lower bound of R\'ev\'esz and Sarantopoulos on the norm of a product of linear functionals on a real Euclidean space (this subject is sometimes called the `real linear polarization constant' problem).