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 semi-metric space (X,d) that is known to have strict

This is an announcement for the paper "Strict p-negative type of a semi-metric space" by Hanfeng Li and Anthony Weston. Abstract: Doust and Weston introduced a new method called "enhanced negative type" for calculating a non trivial lower bound p(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 p(T) p-negative type for some non negative p. This allows us to significantly improve the lower bounds on the supremal strict p-negative type of finite metric trees that were given by Doust and Weston and, moreover, leads in to one of our main results: The supremal p-negative type of a finite semi-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 semi-metric space (finite or otherwise) has p-negative type for some p > 0, then it must have strict q-negative type for all q in [0,p). This generalizes a well known theorem of Schoenberg and leads to further applications. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B20 Remarks: 12 pages The source file(s), HLAW-Final.tex: 44858 bytes, is(are) stored in gzipped form as 0901.0695.gz with size 13kb. The corresponding postcript file has gzipped size 353kb. Submitted from: westona@canisius.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0901.0695 or http://arXiv.org/abs/0901.0695 or by email in unzipped form by transmitting an empty message with subject line uget 0901.0695 or in gzipped form by using subject line get 0901.0695 to: math@arXiv.org.