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We prove that a nonempty closed and geodesically convex subset of the$l_{\infty}$ plane $\mathbb{R}^2_{\infty}$ is hyperconvex and we characterizethe tight spans of arbitrary subsets of $\mathbb{R}^2_{\infty}$ via thisproperty: 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 $X$.

Tight Span of Subsets of The Plane With The Maximum Metric