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 $x,y\in K$

5.
$$

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 the
set 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 $$ then there is a real number $s$ such that $$ and
so $$. Therefore we have that $x=s\frac{x}{s}+(1s)0\in K$, that contradicts to the
fact that $x\in Ex(K).$
Title  properties of Minkowski’s functional 

Canonical name  PropertiesOfMinkowskisFunctional 
Date of creation  20130322 15:45:04 
Last modified on  20130322 15:45:04 
Owner  georgiosl (7242) 
Last modified by  georgiosl (7242) 
Numerical id  10 
Author  georgiosl (7242) 
Entry type  Theorem 
Classification  msc 46B20 