Extremal positive pluriharmonic functions on euclidean balls

Contrary to the well understood structure of positive harmonic functions in the unit disk, most of the properties of positive pluriharmonic functions in symmetric domains of Cⁿ, in particular the unit ball, remain mysterious. In particular, in spite of efforts spread over quite a few decades, no characterization of the extremal rays in the cone of positive pluriharmonic functions in the unit ball of Cⁿ is known. We investigate this question by a geometric tomography technique, and provide some new classes of examples of such extremal functions.