properties of Minkowski’s functional

Let $X$ be a normed space, $K$ convex subset of $X$ and $0$ belongs to the interior of $K$.Then

1. ${\rho }_K(x)\ge 0$ for all $x\in X$

2. ${\rho }_K(0)=0$

3. ${\rho }_K(\lambda x)=\lambda {\rho }_K(x)$, for all $\lambda \ge 0$ and $x\in X$

4. ${\rho }_K(x+y)\le {\rho }_K(x)+{\rho }_K(y)$ for all $x,y\in K$

6. where $K^0$ denotes the interior of $K$

7. $\overline{K}=\{x\in X:{\rho }_K(x)\le 1\}$ where $\overline{K}$ denotes the closure of $K$

8. $Bd(K)=\{x\in X:{\rho }_K(x)=1\}$ where the $Bd(K)$ denotes the boundary of $K$.

Minkowski’s functional is a useful tool to prove propositions and solve exercises. Let us see an example
Example Let $K$ be a convex subset of $X$. Show that $Ex(K)\subset Bd(K)$, where $Ex(K)$ denotes theset of extreme points of $K$.
If $x\in Ex(K)$ then from this follows that $x\in 1K$ and ${\rho }_K(x)=1$.Now we hypothesize that then there is a real number $s$ such that andso . Therefore we have that $x=s\frac{x}{s}+(1-s)0\in K$, that contradicts to thefact that $x\in Ex(K).$