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In relation to this question, I'm trying to solve a more complicated problem of the following. It includes two parameters $\alpha$ and $\beta$ where $\alpha\in(0,1)$ and $\beta\leq 1$.

There are two RV's $X_1$ and $X_2$, independently and uniformly distributed over $[0,1]$.

The problem I'd like to solve is

$$\max_{A\subset[0,1]^2}(1-\beta)\int_A\big(\alpha x_1+(1-\alpha)x_2\big)dx_1dx_2+\beta\int_Adx_1dx_2$$ $$s.t. E[X_1|(X_1,X_2)\in A]\geq y~\textrm{and}\\E[X_2|(X_1,X_2)\in A]\geq y.$$

For a given $y\in(0,1)$. As in the maxim program above, $A$ should be a subset of $[0,1]^2$. For the parameter value $\beta=1$, the problem is reduced to the problem of the above link. However, for any value of $\beta$ smaller than 1, it gives more weights to the first term which is a weighted average of $x_1$ and $x_2$, with the weight being $\alpha$.

I am trying to solve this problem using the same technique used so solve the problem in the link, but it looks like it cannot be directly applied.. Any hints, suggestion or an idea of a solution to the problem?

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