How to prove that the support function of a convex body can be written: $ h(K,u)=(x_0,u)$ with $x_0\in\partial K$

Question

Let be $K\subset\mathbb{R}^n$ a compact convex set.

The support function associated with $K$, $h(K,\cdot):\mathbb{R}^n\rightarrow(-\infty,+\infty)$, is defined in the following way: $$ h(K,u)=\sup_{y\in K}\;(y,u).$$

Since $K$ is compact, we have that the support function is well defined and there exists a $x_0\in K$ such that $$h(K,u)=\max\limits_{y\in K}\;(y,u)=(x_0,u).$$

Considering the geometric meaning of support function, I can see that this $x_0$ must be a boundary point of $K$.

How can I prove analytically that $x_0\in\partial K$?

Thank you!

Answer 1

Let $x_0$ be an interior point of $K$, i.e., there exists $r>0$ such that $B_r(x_0) \subset K$. Let $u\in\mathbb{R}^n$, $u\neq 0$. Then $y := x_0 + r \frac{u}{|u|} \in K$ and $$ (y, u) = (x_0 + r \frac{u}{|u|}, u) = (x_0, u) + r |u| > (x_0, u), $$ hence the maximum in the definition of $h(K,u)$ cannot be achieved at $x_0$.