THE SUPREMAL p-NEGATIVE TYPE OF A FINITE SEMI-METRIC SPACE CANNOT BE STRICT

Doust and Weston [5] introduced a new method called “enhanced negative type” for calculating a non trivial lower bound ℘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 ℘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 semi-metric space (X, d) that is known to have strict p-negative type for some p ≥ 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 [5] and, moreover, leads in to our main result: The supremal p-negative type of a finite semimetric 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 ∈ [0, p). This generalizes Schoenberg [21, Theorem 2].