Strict p-negative type of a metric space

Doust and Weston (J Funct Anal 254:2336-2364, 2008) have 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) > 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 metric space (X, d) that is known to have strict p-negative type for some p a parts per thousand yen 0. This allows us to significantly improve the lower bounds on the supremal strict p-negative type of finite metric trees that were given in Doust and Weston (J Funct Anal 254:2336-2364, 2008, Corollary 5.5) and, moreover, leads in to one of our main results: the supremal p-negative type of a finite 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 metric space (finite or otherwise) has p-negative type for some p > 0, then it must have strict q-negative type for all q is an element of [0, p). This generalizes Schoenberg (Ann Math 38:787-793, 1937, Theorem 2) and leads to a complete classification of the intervals on which a metric space may have strict p-negative type.