Tight Span of Subsets of The Plane With The Maximum Metric

We prove that a nonempty closed and geodesically convex subset of the $l_{\infty}$ plane $\mathbb{R}^2_{\infty}$ is hyperconvex and we characterize the tight spans of arbitrary subsets of $\mathbb{R}^2_{\infty}$ via this property: Given any nonempty $X\subseteq\mathbb{R}^2_{\infty}$, a closed, geodesically convex and minimal subset $Y\subseteq\mathbb{R}^2_{\infty}$ containing $X$ is isometric to the tight span $T(X)$ of $X$.