Tight Span of Subsets of The Plane With The Maximum Metric

TL;DR

This paper characterizes the tight spans of arbitrary subsets in the $l_{ ext{infinity}}$ plane, showing they are isometric to minimal closed, geodesically convex, and hyperconvex subsets containing the original set.

Contribution

It establishes that nonempty closed, geodesically convex subsets of the $l_{ ext{infinity}}$ plane are hyperconvex and characterizes tight spans via these subsets.

Findings

Nonempty closed, geodesically convex subsets are hyperconvex.

Minimal such subsets containing a given set are isometric to its tight span.

Provides a geometric characterization of tight spans in the $l_{ ext{infinity}}$ plane.

Abstract

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 $X$.