Minkowski functional

Let $X$ be a normed space and let$K$ an absorbing convex subset of $X$ such that$0$ is in the interior of $K$.Then theMinkowski functional$\rho :X\to ℝ$ is defined as

$$\rho (x)=inf\{\lambda >0:x\in \lambda K\}.$$

We put $\rho (x)=0$ whenever $x=0$. Clearly $\rho (x)\ge 0$ for all $x$.

It is important to note that in general $\rho (x)\ne \rho (-x)$.

Properties
$\rho$ is positively $1$- homogeneous. This means that

$$\rho (s\cdot x)=s\cdot \rho (x)$$

for $s>0$.