Tight span of subsets of the plane with the maximum metric

We prove that a nonempty closed and geodesically convex subset of the l∞l∞ plane R∞2 is hyperconvex and we characterize the tight spans of arbitrary subsets of R∞2 via this property: Given any nonempty X⊆R∞2, a closed, geodesically convex and minimal subset Y⊆R∞2 containing X   is isometric to the tight span T(X)T(X) of X.