Some notes on preferred point [alpha]-geometry and [alpha]-divergence function

A preferred point [alpha]-geometry is introduced in this note, which is a natural extension of the work of Critchley et al. (1993, 1994). Using this geometry, the homogeneous structure and the duality are discussed. We also define a new [alpha]-divergence function and produce a flatness condition, under which the [alpha]-divergence function agrees with a squared preferred point geodesic distance. Further, the Pythagorean result given by Amari (1990) and Critchley et al. (1994) is extended to the [alpha]-divergence function.