Point-wise Symmetry of Birkhoff-James Orthogonality and Geometry of $\mathbb{B}(\ell_\infty^n,\ell_1^m)$

TL;DR

This paper explores the symmetry properties of Birkhoff-James orthogonality in the space of operators from $ ext{ell}_ ext{infty}^n$ to $ ext{ell}_1^m$, linking these properties to the geometric structure and extreme points of the space.

Contribution

It establishes that non-zero left-symmetric points are smooth and that for $n extgreater 3$, right-symmetric points are extreme points, advancing the understanding of the space's geometry.

Findings

Non-zero left-symmetric points are smooth.

For $n extgreater 3$, right-symmetric points are extreme points.

First step towards characterizing extreme points and computing Grothendieck constants.

Abstract

We study the relationship between the point-wise symmetry of Birkhoff-James orthogonality and the geometry of the space of operators $\mathbb{B}(\ell_\infty^n,\ell_1^m)$. We show that any non-zero left-symmetric point in this space is a smooth point. We also show that for $n\geq4$, any unit norm right-symmetric point of this space is an extreme point of the closed unit ball. This marks the first step towards characterizing the extreme points of these unit balls and finding the Grothendieck constants $G(m,n)$ using Birkhoff-James orthogonality techniques.