Proposition: Cardinal anisotropy

In the cardinal transport model with displacements $\pm a{e}_j$, each of duration $a/c$, and waiting,

For a Euclidean unit direction $u$, its maximal scheduled radial speed is

In two dimensions it is $c$ on an axis and $c/\sqrt{2}$ on a diagonal.

Proof. The cardinal edge velocities are $\pm c{e}_j$. Their convex hull with $0$ is the ${\ell }^1$ ball of radius $c$. Time-sharing cardinal movements and waiting attains every point in that hull asymptotically. A vector $\rho u$ lies in the hull exactly when $\rho \Vert u{\Vert }_1\le c$.