Tight Span of Subsets of The Plane With The Maximum Metric

Abstract

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

22 pages, 23 figures