The sum in the excepted value is bounded below by $n$ (by the arithmetic-geometric mean inequality). The lower bound is achieved when all the variables are equal. Whence the value of you minimum is $n$ and is achieved for the coupling $X_1=\dots=X_n=Y$, $Y\sim \mathrm{U}[0,1]$.

The sum in the excpected value is bounded below by $n$ (by the arithmetic-geometric mean inequality). The lower bound is achieved when all the variables are equal. Whence the value of you minimum is $n$ and is achieved for the coupling $X_1=\dots=X_n=Y$, $Y\sim \mathrm{U}[0,1]$.