  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.
