OPTIMAL LOWER BOUND ON THE SUPREMAL STRICT p-NEGATIVE TYPE OF A FINITE METRIC SPACE

Abstract. Determining meaningful lower bounds on the supremal strict p-negative type of classes of finite metric spaces is a difficult nonlinear problem. In this paper we use an elementary approach to obtain the following result: Given a finite metric space (X, d) there is a constant ζ> 0, dependent only on n = |X | and the scaled diameter D = (diam X)/min{d(x, y)|x 6 = y} of X (which we may assume is> 1), such that (X, d) has p-negative type for all p ∈ [0, ζ] and strict p-negative type for all p ∈ [0, ζ). In fact, we obtain ζ = l