  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) > 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 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 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 in [0,p). This
generalizes a well known theorem of Schoenberg and leads to a complete
classification of the intervals on which a metric space may have strict
p-negative type. (Several of the results in this paper hold more generally for
semi-metric spaces.)
