Properties of the Minkowski Gauge

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $X$ be a real normed space with zero vector $0_{X}$, let $U\subseteq X$ be open and convex with $0_{X}\in U$, and let $p_{U}$ be its gauge.

(i) Sublinearity. $p_{U}$ is a sublinear functional on $X$.

(ii) Bounds. If $r>0$ is a real number with $B(0_{X},r)\subseteq U$, then

$$0\le p_{U}(x)\le r^{-1}\lVert x\rVert\qquad\text{for every }x\in X.$$

(iii) The set as a sublevel set.

$$U=\{x\in X:p_{U}(x)<1\}.$$