Enhanced negative type for finite metric trees

A finite metric tree is a finite connected graph that has no cycles, endowed with an edge weighted path metric. Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities (1) that encode the best possible quantification of the strictness of the non trivial 1-negative type inequalities for finite metric trees. These inequalities are sufficiently strong to imply that any given finite metric tree (T, d) must have strict p-negative type for all p in an open interval (1 − ζ, 1 + ζ), where ζ > 0 may be chosen so as to depend only upon the unordered distribution of edge weights that determine the path metric d on T . In particular, if the edges of the tree are not weighted, then it follows that ζ depends only upon the number of vertices in the tree. We also give an example of an infinite metric tree that has strict 1-negative type but does not have p-negative type for any p > 1. This shows that the maximal p-negative type of a metric space can be strict.