Joint numerical radius of Tuples: Extreme points, subdifferential set and Gateaux derivative

Suppose $\mathcal{Z}$ is the space of all tuples of operators on a finite-dimensional Banach space endowed with the joint numerical radius norm. We obtain the structure of the extreme points of the dual unit ball of $\mathcal{Z}\mathrm{.}$ Using this, we derive an expression for the subdifferential set of the joint numerical radius of a tuple in $\mathcal{Z}\mathrm{.}$ Applying this expression, we characterize smooth tuples and Birkhoff-James orthogonality in $\mathcal{Z}\mathrm{.}$ Finally, we obtain the Gateaux derivative of the joint numerical radius of a tuple.