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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.

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    $\begingroup$ Put $X=e^Y$. By Jensen's inequality, $E e^Y\ge e^{EY}=e^C$. Since concentrating $X$ at $e^C$ achieves that bound, it is the best. $\endgroup$
    – Brendan McKay
    Commented May 27, 2020 at 3:47

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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 $\mathbb{P}(X=e^C)=1$.

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    $\begingroup$ I beat you by 3 minutes, but I hope we didn't just do someone's homework. $\endgroup$
    – Brendan McKay
    Commented May 27, 2020 at 4:24

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