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

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 = ln ( 1/(1− 0) ) ln D where 0 = 1 2 ( 1 bn/2c + 1 dn/2e ) . A consideration of basic examples shows that our value of ζ is optimal provided that D≤ 2. In other words, for each D ∈ (1, 2] and natural number n ≥ 3, there exists an n-point metric space of scaled diameter D whose supremal strict p-negative type is exactly ζ . The results of this paper hold more generally for all finite semi-metric spaces since the triangle inequality is not used in any of the proofs. Moreover, ζ is always optimal in the case of finite semi-metric spaces. 2000 Mathematics subject classification: primary 46B20.