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

The tight span of a finite metric space (X,d) is the metric space T(X,d) consisting of the compact faces of the polytopeP(X,d)≔{f∈RX:f(x)+f(y)⩾d(x,y) for all x,y∈X},endowed with the metric induced by the l∞-norm on RX. In this paper, we study T(X,d) in case d is antipodal i.e., in case there is a map σ:X→2X-{∅} with d(x,y)+d(y,z)=d(x,z) holding for all x,y∈X and z∈σ(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).