Metrics defined by Bregman Divergences

Bregman divergences are generalizations of the well known Kullback-Leibler divergence.They are based on convex functions and have recently received great attention.We present a class of "squared root metrics" based on Bregman divergences.They can be regarded as natural generalization of Euclidean distance.We provide necessary and sufficient conditions for a convex function so that the square root of its associated average Bregman divergence is a metric.