Supporting hyperplane

We say that is a supporting hyperplane to function at if:

for all .

In a sense is always “above” the plane .

Theorem (Convexity)

is convex
if and only if
there exists a supporting hyperplane at every .

Theorem (Gradient)

If is differentiable at
and if has a supporting hyperplane at ,
then .

Proof

If is a supporting hyperplane,
then for any and we have

Taking we get for any .
So it has to be that .