*Ian
Doust's *i*nformation pages*

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Enhanced negative type for finite metric trees

*J. Funct. Anal.,* **254** (2008), 2336-2364.
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Ian Doust and Anthony Weston

**Address:** (ID) School of Mathematics and Statistics, University of New South Wales, Sydney
NSW 2052, Australia
**Email:** i.doust@unsw.edu.au

**Address:** (AW) Department of Mathematics and Statistics, Canisius College,
Buffalo, New York 14208, United States of America

**Email:** westona@canisius.edu

**Abstract:**
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 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.

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