Minkowski functional

Remarks

Since $K$ is absorbing, the set $\set{t > 0 : x \in tK}$ is nonempty, so $p_K(x)$ is a finite number. If $K$ is convex, then $p_K$ is sublinear: $p_K(x + y) \le p_K(x) + p_K(y)$ and $p_K(s x) = s \, p_K(x)$ for $s \ge 0$. If $K$ is convex and balanced, then $p_K$ is a seminorm. Example: let $\norm{\cdot}$ be a norm on $V$ and $K = \set{x : \norm{x} \le 1}$; then $x \in tK$ if and only if $\norm{x} \le t$, so $p_K(x) = \norm{x}$. Non-example: in $\R^2$ the segment $K = [-1, 1] \times \set{0}$ is not absorbing, and for $x = (0, 1)$ the set $\set{t > 0 : x \in tK}$ is empty, so no finite value $p_K(x)$ is defined. Some authors allow any set $K$ and set $p_K(x) = \infty$ when no $t$ exists. In a topological vector space, the Minkowski functional of a convex open neighbourhood $K$ of $0$ is continuous, and $K = \set{x : p_K(x) < 1}$.