Definition 11.1.11 (Gauge/Minkowski Functional).label Let $E$ be a vector space over $K \in \RC$ and $A \subset E$ be radial, then the mapping
is the gauge/Minkowski functional of $A$, and
- (1)
For any $x \in E$ and $\lambda \ge 0$, $[\lambda x]_{A} = \lambda [x]_{A}$.
- (2)
If $A$ is convex, then for any $x, y \in E$, $[x + y]_{A} \le [x]_{A} + [y]_{A}$.
- (3)
If $A$ is circled, then for any $x \in E$ and $\lambda \in K$, $[\lambda x]_{A} = \abs{\lambda}[x]_{A}$.
- (4)
If $A$ is circled, then $\bracs{\rho < 1}\subseteq A \subseteq \bracs{\rho \le 1}\subseteq \ol A$.
In particular,
- (5)
If $A$ is convex, then $[\cdot]_{A}$ is a sublinear functional.
- (6)
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 $[x + y]_{A} \le [x]_{A} + [y]_{A}$.
(4): Let $x \in \bracs{\rho \le 1}$, then $\lambda x \in A$ for all $\lambda \in (0, 1)$. Therefore
so $x \in \overline{A}$.$\square$
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