Metricity preserving transforms

Abstract Given a metric d on some set A and a transformation φ: R + → R + it is often required to determine the metricity of the transformed function D = φ ( d ) on A . Though in general this needs a thorough investigation of both φ and d , in a number of cases the metricity of D is trivially implied by the metricity of d . Such transformations, for which φ ( d ) is a metric whenever d is a metric (the choice of this d and the underlying A is arbitrary), are called Metricity Preserving Transforms (MPT). In this paper we present a necessary and sufficient condition for φ to be metricity preserving. A generalization is also provided for transforms composing a number of metrics. Suitable examples are cited for illustration.