Let $n \in \mathbb{N}$, and let $U,V \subset \mathbb{R}^n$ be open domains equipped with their Borel $\sigma$-algebras, with $U$ convex. Let $\rho_0:U \to (0,\infty)$ and $\rho_1:V \to (0,\infty)$ be continuous functions satisfying
Let $\phi:U \to \mathbb{R}$ be a convex function of class $C^2(U)$, and define the Borel map $T:U \to V$ by $T(x)=\nabla \phi(x)$. Assume that $T(U)$ is open in $\mathbb{R}^n$, that $T(U) \subset V$, and that the corestriction $T:U \to T(U)$ is a $C^1$ diffeomorphism. Define finite Borel measures $\mu$ on $U$ and $\nu$ on $V$ by