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Let $L$ be a big holomorphic line bundle on a complex projective manifold$X.$ We show how to associate a convex function on the Okounkov body of $L$ toany continuous metric $\psi$ on $L.$ We will call this the Chebyshev transformof $\psi,$ denoted by $c[\psi].$ Our main theorem states that the difference ofmetric volume of $L$ with respect to two metrics, a notion introduced byBerman-Boucksom, is equal to the integral over the Okounkov body of thedifference of the Chebyshev transforms of the metrics. When the metrics havepositive curvature the metric volume coincides with the Monge-Amp\`ere energy,which is a well-known functional in K\"ahler-Einstein geometry and Arakelovgeometry. We show that this can be seen as a generalization of classicalresults on Chebyshev constants and the Legendre transform of invariant metricson toric manifolds. As an application we prove the differentiability of themetric volume in the cone of big metrized $\mathbb{R}$-divisors. Thisgeneralizes the result of Boucksom-Favre-Jonsson on the differentiability ofthe ordinary volume of big $\mathbb{R}$-divisors and the result ofBerman-Boucksom on the differentiability of the metric volume when theunderlying line bundle is fixed.

Transforming metrics on a line bundle to the Okounkov body