  We study the Steklov eigenvalue problem for the $\infty-$orthotropic Laplace
operator defined on convex sets of $\mathbb{R}^N$, with $N\geq2$, considering
the limit for $p\to+\infty$ of the Steklov problem for the $p-$orthotropic
Laplacian. We find a limit problem that is satisfied in the viscosity sense and
a geometric characterization of the first non trivial eigenvalue. Moreover, we
prove Brock-Weinstock and Weinstock type inequalities among convex sets,
stating that the ball in a suitable norm maximizes the first non trivial
eigenvalue for the Steklov $\infty-$orthotropic Laplacian, once we fix the
volume or the anisotropic perimeter.
