To the geometry of spaces of plurisubharmonic functions on a K\"ahler manifold

Consider a compact K"ahler manifold $(X,\omega )$ and the space $\cal E(X,\omega)=\cal E$of$\omega$–plurisubharmonic functions of full Monge–Amp`ere mass on it. We introduce a quantity $\rho [u,v]$ to measure the distance between $u,v\in \mathcal{E}$; $\rho [u,v]$ is not a number but rather a decreasing function on a certain interval $(0,V)\subset \mathbb{R}$. We explore properties of $\rho [u,v]$, and using them we study Lagrangians and associated energy spaces of $\omega$–plurisubharmonic functions. Many results here generalize Darvas’s findings about his metrics $d_{\chi }$.