Definition 9.1.9 (Gauge/Minkowski Functional). Let $E$ be a vector space over $K \in \RC$ and $A \subset E$ be a radial set, then the mapping
is the gauge/Minkowski functional of $A$, and
For any $x \in E$ and $\lambda \ge 0$, $[\lambda x]_{A} = \lambda [x]_{A}$.
If $A$ is convex, then for any $x, y \in E$, $[x + y]_{A} \le [x]_{A} + [y]_{A}$.
If $A$ is circled, then for any $x \in E$ and $\lambda \in K$, $[\lambda x]_{A} = \abs{\lambda}[x]_{A}$.
In particular,
If $A$ is convex, then $[\cdot]_{A}$ is a sublinear functional.
If $A$ is convex and circled, then $[\cdot]_{A}$ is a seminorm.
Proof. (2): Let $\lambda, \mu > 0$ such that $\lambda^{-1}x, \mu^{-1}y \in A$. By convexity, $t\lambda^{-1}+ (1 - t)\mu^{-1}y \in A$ for all $t \in [0, 1]$. Let $t \in [0, 1]$ such that
then $(\lambda + \mu)^{-1}\in A$, and $\lambda + \mu \ge [x + y]_{A}$. Thus $[x + y]_{A} \le [x]_{A} + [y]_{A}$.$\square$