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 $s > 0$ .