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 the Minkowski functional $\rho\colon X\to\mathbb{R}$ is defined as

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

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

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

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

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

for $s>0$ .