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.

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 #1

References in corpus (1)

• Metric Cotype

Cited by in corpus (8)

• Positive definite metric spaces
• Metric trees of generalized roundness one
• Polygonal equalities and $p$-negative type
• Beyond trees: the metric geometry of subsets of weighted Hamming cubes
• Roundness Properties of Banach spaces
• Roundness properties of ultrametric spaces
• Optimal lower bounds on the maximal p-negative type of finite metric spaces
• Strict p-negative type of a metric space