What is the name of this theorem? convex set

Question

I have seen this theory but I am wondering what is the name of this theorem?

If

$~~~~~~$ 1- $\mathbb D$ is a closed convex set and real

$~~~~~~$ 2- $f:\mathbb D \rightarrow\mathbb R$ is a convex function,

$~~~~~~$ 3- $f$ has a maximum over $\mathbb D$

Then

$~~~~~~$ The maximum of $f$ is at an extreme point of $\mathbb D$.

Answer 1

This is called the Maximum Principle for convex functions. (See the final sentence of the introduction.)

If a maximum occurred at an interior point, then any line segment passing through the point would have ends at lower value than the middle point where the maximum occurs, violating convexity.