Let $X= (X\_1,\dots, X\_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set.


\begin{equation}\label{main12}
C= \{x\in \mathbb{R}^d ~|~ h(x)\leq \mathop{\sup}\_{Q\in \mathcal{D}}\mathbb{E}\_{Q}(h(X)),\qquad \forall h\in \mathcal{A}\},
\end{equation}


where $\mathbb{E}\_{Q}$ is the expectation with respect to a measure $Q$ (integration in measure theory language), $\mathcal{D}$ is a set of probability measures and $\mathcal{A} = \{h~|~ h: \mathbb{R}^d \rightarrow \mathbb{R},\qquad h \hbox{ is a non-linear convex function}\}$.


Can we characterize the set $C$? Or, at least can we say something about how to approximate the elements of $C$?



Characterization of a set in $\mathbb{R}^d$
Suppose $x\in C$. Then we can make $h\_n$ closer and closer to the projection $g$ given by $g(x)=x\_i$, and $\hat h\_n$ closer and closer to $\hat g(x)=-x\_i$, so upon taking limits, $x\_i\le \sup\mathbb E\_Q(X\_i)$ and $-x\_i\le\sup \mathbb E\_Q(-X\_i)=-\inf \mathbb E\_Q(X\_i)$.


(May need to assume $\mathcal D$ is nice enough that the sup's are actually max's here.)


Then by Jensen's inequality,
$$
\{E\_Q(X): Q\in \mathcal D\}\subseteq C\subseteq \{x: x\_i\in [\inf\_Q\mathbb E\_Q(X\_i), \sup\_Q\mathbb E\_Q(X\_i)]\quad\forall i\}.
$$


