Transforming metrics on a line bundle to the Okounkov body

Abstract

Let $L$ be a big holomorphic line bundle on a compact complex manifold $X.$ We show how to associate a convex function on the Okounkov body of $L$ to any continuous metric $e^{-\psi}$ on $L.$ We will call this the Chebyshev transform of $\psi,$ denoted by $c[\psi].$ Our main theorem states that the integral of the difference of the Chebyshev transforms of two weights is equal to the relative energy of the weights, which is a well-known functional in K\ahler-Einstein geometry and Arakelov geometry. We show that this can be seen as a generalization of classical results on Chebyshev constants and the Legendre transform of invariant metrics on toric manifolds. As an application we prove the differentiability of the relative energy in the ample cone.