On the Geometry of Metrics Embeddable in the Real Line

On a fixed finite set {1, . . . , n}, we consider the set of metrics for which the metric space can be isometrically embedded in the real line. To understand the geometry of this set, we study its convex hull, Qn, and the closure of its convex hull, Qn. We first show how the set of metrics is contained in its convex hull and characterize all unbounded one-dimensional extreme subsets of Qn combinatorially. Secondly we give a combinatorial characterization of the set of unbounded edges of Qn. As a simple by-product, we obtain that Qn is closed if and only if n ≤ 3. Apart from being of interest in its own, these metrics have links to the socalled Linear Arrangement Problem of combinatorial optimization and in particular to the so-called spreading metrics which have been used for approximation algorithms.