Maximality with respect to inclusion

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

and .

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.