Polygonal equalities and p-negative type

Nontrivial p-polygonal equalities impose certain conditions on the geometry of a metric space $(X,d)$ and so it is of interest to be able to identify the values of p ∈ [0,∞) for which such equalities exist. Following work of Li and Weston, Kelleher, Miller, Osborn and Weston established that if a metric space $(X,d)$ is of p-negative type, then $(X,d)$ admits no nontrivial p-polygonal equalities if and only if it is of strict p-negative type. In this note we remove the underlying premise of p-negative type from this theorem. As an application we show that the set of all p for which a finite metric space $(X,d)$ admits a nontrivial p-polygonal equality is always a closed interval of the form [, ∞), where > 0, or the empty set. It follows that for each q = 2, the Schatten q-class Cq admits a nontrivial p-polygonal equality for each $p > 0$. Other spaces with this same property include $C[0, 1]$ and q⁽³⁾ for all $q > 2$.