Given a set $R = (R_1, R_2, ..R_n)$ with $n$ positive random variables.
The sum is a fixed constant $C$.
What's the $E(min(R)+ max(R))$ ?
$R_1, ..R_{n-1}$ is uniform distributed from $(0, \frac{C}{n})$. $R_n$ is just an offset.
The $n-1$ variables are not necessary IID. There are some $k$ degrees of freedom among the $n-1$, and $k<n-1$. Assume the covariance matric is $\Sigma^{n-1 \times n-1}$, with $rank(\Sigma) =k$
I suppose that you extract only $R_1...R_{n-1}$ and then $R_n$ is uniquely determined as a function of the $R_1...R_{n-1}$ values as $R_n = c-R_1-...-R_{n-1}$.
In this case you have to compute
$$ E_n = \int p(x_1,...,x_{n-1})[\min(x_1,...,x_{n-1}, c-x_{1}-...-x_{n-1})+ \\ +\max(x_1,...,x_{n-1}, c-x_{1}-...-x_{n-1})] dx_1...dx_{n-1} $$
where the integral is over the domain $[0,c/n]^{n-1}$.
For $c>0$, if the variables are i.i.d. over $[0,c/n]^{n-1}$, I obtain (using integration software) $$ E_2 = c/4, \quad E_3 = 7c/729, \quad E_4 = 11c/65536, \quad E_5 = 16c/9765625 $$ ...it seems it is going to zero quite quickly. Clearly a very partial answer but I hope it will give some insight.