Enhanced negative type for finite metric trees

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 #

Additive combination spaces

“…In particular we deduce a formula for the p-negative type gap of the space in terms of the p-negative type gaps of the components, independent of how the components are arranged in the ambient space. This generalizes earlier work on metric trees by Doust and Weston [DW08b,DW08a]. The results hold for semi-metric spaces as well, as the triangle inequality is not used.…”

“…Our proof of Theorem 1.4 does not work with p-negative type directly, but an equivalent property known as generalized roundness p. Enflo [Enf69] introduced the ideas of roundness and generalized roundness to answer in the negative a question of Smirnov's: "Is every separable metric space uniformly homeomorphic to a subset of L 2 [0, 1]?" In 1997 Lennard, Tonge and Weston [LTW97] showed that the notions of negative type and generalized roundness coincide: a metric space (X, d) has p-negative type if and only if it has generalized roundness p. The notion of strict generalized roundness p was formalized by Doust and Weston in [DW08b], and shown to be equivalent to strict p-negative type.…”

“…The definition of Γ p X given above is not the original form in which it was defined in [DW08b,DW08a], and its translation into the p-negative type setting gives the awkward scaling factor above. We will in fact work with its original incarnation, which we come to in Section 3.…”

The geometry of two-valued subsets of Lp -spaces

“…Theorem 2.1 is a variation of themes developed in Weston [16], Lennard et al [9], and Doust and Weston [4]. The motivation for all such studies has stemmed from Enflo's original definition of generalized roundness that was given in [5].…”

Estimating the gap of finite metric spaces of strict p-negative type

“…The above defined p-negative type gap Γ(X, p) can be used to enlarge the p-parameter, for wich a given finite metric space is of p-negative type: It is shown in [9] (Theorem 3.3) that a finite metric space (X, d) with cardinality n = |X| ≥ 3 of strict p-negative type is of strict q-negative type for all q ∈ For basic information on p-negative type spaces (1-negative type spaces are also known as quasihypermetric spaces) see for example [2,8,9,12,13,14,15,16,18].…”