Let $C$ be a non-empty convex set of $\mathbb{R}^n$. The gauge function of $C$ is defined as $\gamma(x|C) = \inf\left\{\lambda \ge 0 | x \in \lambda C \right\}$ (Rockafellar, Convex Analysis). According to my intuition, $\gamma(\cdot |C)$ is in general different thatn $\gamma(\cdot|\operatorname{cl}(C))$, where $\operatorname{cl}(C)$ is the closure of $C$. But is it true that the closures of these two gauge functions coincide? (The epigraph of the closure of a proper convex function is the closure of the epigraph of the function, see Rockafellar).
[Math] gauge function of convex set and closures
convex-analysis
Best Answer
The inequality $\gamma(\cdot | C) \ge \gamma(\cdot | \operatorname{cl}(C))$ implies $$\operatorname{epi}(\gamma(\cdot | C)) \subset \operatorname{epi}(\gamma(\cdot | \operatorname{cl}(C))).$$ It remains to prove $$\operatorname{cl}(\operatorname{epi}(\gamma(\cdot | C))) \supset \operatorname{epi}(\gamma(\cdot | \operatorname{cl}(C))).$$ Let $\alpha = \gamma(x | \operatorname{cl}(C))$. Since $\operatorname{cl}(C)$ is closed, we have $x \in \alpha \, \operatorname{cl}(C)$, i.e., there is a sequence $\{c_n\}_{n \in \mathbb N} \subset C$ with $c_n \to c$ and $x = \alpha \, c$. Thus, $(\alpha, \alpha \, c_n) \in \operatorname{epi}(\gamma(\cdot | C))$ and $(\alpha, \alpha \, c_n) \to (\alpha, x) \in \operatorname{cl}(\operatorname{epi}(\gamma(\cdot | C)))$. This provides $(\beta, x) \in \operatorname{cl}(\operatorname{epi}(\gamma(\cdot | C)))$ for all $(\beta, x) \in \operatorname{epi}(\gamma(\cdot|\operatorname{cl}(C)))$.