Continuity and Level Sets of the Minkowski Gauge

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

Let $X$ be a normed vector space and let $\Omega\subset X$ be convex with $0\in\operatorname{int}(\Omega)$ (see interior).

Corollary: The Minkowski gauge $p_\Omega$ is continuous. Moreover,

\operatorname{int}(\Omega)=\{x\in X\mid p_\Omega(x)<1\}

and

\overline{\Omega}=\{x\in X\mid p_\Omega(x)\le 1\}.

Here $\overline{\Omega}$ denotes the closure of $\Omega$ .

Remarks

Context: Continuity comes from the inclusion of a norm ball into $\Omega$ (since $0\in\operatorname{int}(\Omega)$ ), yielding a Lipschitz bound for $p_\Omega$ . The set identities combine the gauge level-set theorem with lin(Ω)=cl(Ω) for solid convex sets.