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 .