Diameter of Kähler currents

We establish upper bounds on the diameter of compact Kähler manifolds endowed with Kähler metrics whose volume form satisfies an Orlicz integrability condition. Our results extend previous estimates due to Fu-Guo-Song, Y.Li, and Guo-Phong-Song-Sturm. In particular, they do not involve any constraint on the vanishing of the volume form. Moreover, we show that singular Kähler-Einstein currents have finite diameter, provided that their local potentials are Hölder continuous.