convex function boundary points

Question

Consider a convex subset $D \subset \mathbb{R}^n$. and convex functions $f, g : D \rightarrow \mathbb{R}$.

Suppose that $f$ is strictly convex and that $f$ has a maximum at $x^∗$. Show that $x^∗$ must be a boundary point.

I don´t really know how to approach this problem, I would be very thankful for every hint.

Answer 1

Hint:

Suppose $x^*$ is in the interior, draw a small ball, $B$, around $x^*$ such that it is a subset of $D$. Pick two points such that $y,z \in B$ and $x^*=\frac{1}{2}(y+z)$

Apply the definition of strictly convex and the fact that $x^*$ is a maximum to get a contradiction.