  We introduce a novel family of distances, called the chord gap divergences,
that generalizes the Jensen divergences (also called the Burbea-Rao distances),
and study its properties. It follows a generalization of the celebrated
statistical Bhattacharyya distance that is frequently met in applications. We
report an iterative concave-convex procedure for computing centroids, and
analyze the performance of the $k$-means++ clustering with respect to that new
dissimilarity measure by introducing the Taylor-Lagrange remainder form of the
skew Jensen divergences.
