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Suppose that we have a probability measure $\rho$ which is supported on $\mathbb{R}^d$ and absolutely continuous w.r.t. the Lebesgue measure. Take two vector fields $F, G : \mathbb{R}^d \rightarrow \mathbb{R}^d$ and define $\mu = F_{\#\rho}$ and $\nu = G_{\#\rho}$.

Question: What condition on $F$ and $G$ can we impose to get the convex ordering $\mu \leq_{\mathrm{cvx}} \nu$?

Reminder 1: The push-forward is defined as: for any Borel set $A$, $\mu(A) = F_{\#\rho}(A) = \rho(F^{-1}(A))$.

Reminder 2: $\mu \leq_{\mathrm{cvx}} \nu$ i.f.f. for any convex function $h : \mathbb{R}^d \rightarrow \mathbb{R}$, $\mathbb{E}_{X \sim \mu} [h(X)] \leq \mathbb{E}_{Y \sim \nu}[h(Y)]$.

I tried to derive a condition using the fact that for any convex function $h : \mathbb{R}^d \rightarrow \mathbb{R}$, we should have $\mathbb{E}_{X \sim \rho} [h \circ F(X)] \leq \mathbb{E}_{Y \sim \rho}[h \circ G(Y)]$, but I'm not making any progress...

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