Tight Span of Subsets of The Plane With The Maximum Metric

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