Convex function
such that convex, and
for and all .
If is differentiable on , then the following statements are equivalent:
- (a) is convex
- (b) for all
- (c) for all
The equivalence proof of (a), (b), and (c) can be found in https://www.math.washington.edu/~burke/crs/516/notes/ch1.pdf. Also I keep a local copy.
In the following, I am going to proof how to derive (b) from (a).
Proof
From the definition of convex function above, we have
which can be rewritten as
Then we take on both sides, we have
which is equal to (b) .