No abstract

Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities that quantify the extent of the "strictness" of the 1-negative type inequalities for finite metric trees. These inequalities of "enhanced 1-negative type" are sufficiently strong to imply that any given finite metric tree must have strict p-negative type for all values of p in an open interval that contains the number 1. Moreover, these open intervals can be characterized purely in terms of the unordered distribution of edge weights that determine the path metric on the particular tree, and are therefore largely independent of the tree's internal geometry. From these calculations we are able to extract a new non linear technique for improving lower bounds on the maximal p-negative type of certain finite metric spaces. Some pathological examples are also considered in order to stress certain technical points.Comment: 35 pages, no figures. This is the final version of this paper sans diagrams. Please note the corrected statement of Theorem 4.16 (and hence inequality (1)). A scaling factor was omitted in Version #

Abstract. Doust and Weston [8] have introduced a new method called enhanced negative type for calculating a non-trivial lower bound ℘ T on the supremal strict p-negative type of any given finite metric tree (T, d). In the context of finite metric trees any such lower bound ℘ T > 1 is deemed to be non-trivial. In this paper we refine the technique of enhanced negative type and show how it may be applied more generally to any finite metric space (X, d) that is known to have strict p-negative type for some p ≥ 0. This allows us to significantly improve the lower bounds on the supremal strict p-negative type of finite metric trees that were given in [8, Corollary 5.5] and, moreover, leads in to one of our main results: The supremal p-negative type of a finite metric space cannot be strict. By way of application we are then able to exhibit large classes of finite metric spaces (such as finite isometric subspaces of Hadamard manifolds) that must have strict p-negative type for some p > 1. We also show that if a metric space (finite or otherwise) has p-negative type for some p > 0, then it must have strict q-negative type for all q ∈ [0, p). This generalizes Schoenberg [27, Theorem 2] and leads to a complete classification of the intervals on which a metric space may have strict p-negative type.

Abstract. Motivated by a classical theorem of Schoenberg we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space R n of dimension n but it cannot be isometrically embedded in any Euclidean space R r of dimension r < n. We use this result as a technical tool to study "roundness" properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X, d):(1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0, and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative by (4) we thus obtain a short new proof of Lemin's theorem: Every n + 1 point ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class M of all finite metric spaces that may be isometrically embedded into ℓ 2 as an affinely independent set. The results of this paper show that Shkarin's class M consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞].

No abstract