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Enhanced negative type for finite metric trees

J. Funct. Anal., 254 (2008), 2336-2364.

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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