Does the following problem have a solution?
$$
\min\_X \mathbb{E} X
\quad\text{subject to}\quad
\mathbb{E} \log X = C.
$$


Here, the minimization is with respect to all integrable random variables $X$ and $C$ is some constant. Alternatively, instead of minimizing over random variables, one may equivalently view this as optimizing over the space of probability measures.



Minimum mean over all random variables subject to logarithm constraint
Assume $X>0$ a.s. (so the constraint can be satisfied) and write $Y=\log X$. 
By Jensen's inequality (<https://en.wikipedia.org/wiki/Jensen%27s_inequality>), 
$\mathbb{E} X \ge e^{\mathbb{E} Y}=e^C$, so the minimum is attained for $X$ such that $\mathbb{P}(X=e^C)=1$.


