The finite gauge of an open convex neighbourhood of zero

Let $X$ be a real or complex normed space and let $U\subseteq X$ be open and convex with $0\in U$, using Convex sets and continuous real-hyperplane separation in a normed space. For $x\in X$ set $S_x=\{t\in \mathbb{R}:t>0,x\mathrm{/}t\in U\},p_U(x)=\inf S_x\mathrm{.}$ The function $p_U:X\to \mathbb{R}$ is the gauge of $U$, with real nonnegative values and real positive scale parameters. Symmetry and boundedness of $U$ are not assumed.

This infimum is well-defined without HB or choice. Openness at zero gives one $r>0$ with $B(0,r)\subseteq U$. For any fixed $x$ and any $t>\mathrm{\parallel }x\mathrm{\parallel }\mathrm{/}r$, $\mathrm{\parallel }x\mathrm{/}t\mathrm{\parallel }<r$, so $t\in S_x$. Thus $S_x$ is nonempty, for example at $t=1+\mathrm{\parallel }x\mathrm{\parallel }\mathrm{/}r$, and is bounded below by zero. The real infimum property Every nonempty set bounded below has an infimum supplies a finite $p_U(x)\ge 0$. These formulas define a unique value at every $x$; no family of choices is involved. At zero, $S_0=(0,\mathrm{\infty })$, whose infimum is zero because it has members below every positive number.

The scale sets give the following direct calculations. For $U=B(0,1)$, one has $S_x=\{t>0:t>\mathrm{\parallel }x\mathrm{\parallel }\}$, so $p_U(x)=\mathrm{\parallel }x\mathrm{\parallel }$, including zero. For the open convex strip $U=\{(s,v)\in {\mathbb{R}}^2:\mathrm{\mid }s\mathrm{\mid }<1\}$, the condition $(s,v)\mathrm{/}t\in U$ is exactly $t>\mathrm{\mid }s\mathrm{\mid }$, so $p_U(s,v)=\mathrm{\mid }s\mathrm{\mid }$. This set contains $(0,v)$ for all real $v$ and is unbounded; its gauge vanishes along that whole line. Finally, $U=\{0\}$ in a nonzero normed space is convex but not a neighbourhood of zero: every positive-radius ball contains a nonzero multiple of any fixed nonzero vector. For $x\mathrm{\ne }0$ its scale set is empty since $x\mathrm{/}t\mathrm{\ne }0$ for every $t>0$. Thus that set does not define a finite gauge on all of $X$ by this construction.

Brezis Lemma 1.2 and (8), p.6; Teschl (5.1) and Lemma 5.1, pp.137–138.