The tight span of an antipodal metric space—Part I:

Abstract

The tight span of a finite metric space $(X,d)$ is the metric space $T(X,d)$ consisting of the compact faces of the polytope

$$P(X,d)≔\{f\in {\mathbb{R}}^X:f(x)+f(y)⩾d(x,y)\text{for all}x,y\in X\}\text{,}$$

endowed with the metric induced by the $l_{\infty }$-norm on ${\mathbb{R}}^X$. In this paper, we study $T(X,d)$ in case d is antipodal i.e., in case there is a map $\sigma :X\to 2^X-\{\varnothing \}$ with $d(x,y)+d(y,z)=d(x,z)$ holding for all $x,y\in X$ and $z\in \sigma (x)$. In particular, we derive combinatorial results concerning the polytopal structure of the tight span of an antipodal metric space, proving that $T(X,d)$ has a unique maximal cell (i.e. a cell containing all other cells) if and only if $(X,d)$ is antipodal, and that in this case there is a bijection between the facets of $T(X,d)$ and the edges in the so-called underlying graph of $(X,d)$.