Maximum of a Convex Function

Let be a Compact Convex Set.
Let be a Continuous Convex Function.
Then achieves its maximum at an Extreme Point of .

Proof

Let .
It exists as is Compact and is Continuous
Now set

As is Continuous, is open (in )
As is a Convex Function, is a Convex Set.
Suppose contains all Extreme Points of

TODO
Somewhere here, there is a proof that , but I can’t find it rn.