The unit ball in the sup norm is, as everybody else says, a cube. In dimension 3, the boundary is geometrically a flat manifold with singularities at the corners, so the easiest way I can see to compute the distance between any two points is to unfold the faces of the boundary and just draw a straight line segment between the two points. It should be straightforward to work out formulas for all the possibilities. The two points must be on the same face, on adjacent faces, or on opposite faces.

I suspect that something similar can be done in higher dimensions, but I am less confident of the details.

The unit ball in the sup norm is, as everybody else says, a cube. In dimension 3, the boundary is geometrically a flat manifold with singularities at the corners, so the easiest way I can see to compute the distance between any two points is to unfold the faces of the boundary and just draw a straight line segment between the two points. It should be straightforward to work out formulas for all the possibilities. The two points must be on the same face, on adjacent faces, or on opposite faces.

I suspect that something similar can be done in higher dimensions, but I am less confident of the details.