Let where are continuous concave functions and
let ; and .
We define a "special" point to be a point for which the set is maximal, i.e., there is no point for which is properly contained in and is properly contained in .
Assume is the minimizer of (i.e. for any ).
We show there exists a "special point" such that
I am trying to prove by contradiction. So assume there is no "special" point for which . Let be a "special point". Hence, there exist an such that .
So we have
I think the contradiction comes from the maximality of . But I am not sure how to prove this.
I also think that the part with can be proven in a similar way, but I am not sure how.