The Mabuchi geometry of low energy classes

Let $(X,\omega)$ be a K\"ahler manifold and $\psi: \Bbb R \to \Bbb R_+$ be a concave weight. We show that $\mathcal H_\omega$ admits a natural metric $d_\psi$ whose completion is the low energy space $\mathcal E_\psi$, introduced by Guedj-Zeriahi. As $d_\psi$ is not induced by a Finsler metric, the main difficulty is to show that the triangle inequality holds. We study properties of the resulting complete metric space $(\mathcal E_\psi,d_\psi)$.