0
$\begingroup$

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).

  • 0
    Out of curiosity, are you reading Rockafellar because you're interested in convex optimization theory, or do you have other motivations? Is it common for pure mathematicians to read Rockafellar? – 2017-01-12

1 Answers 1

1

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 $(\beta, x) \in \operatorname{epi}(\gamma(\cdot|\operatorname{cl}(C)))$.

  • 0
    I have reversed the first inequality. I think that $x \in \alpha \, cl(C)$ follows since 1. $cl(C)$ is closed and 2. $C$ is convex with $0 \in C$. – 2017-01-12
  • 0
    For defining a gauge function, one typically assumes $0 \in C$, see, e.g., https://en.wikipedia.org/wiki/Minkowski_functional#Definition. – 2017-01-12
  • 0
    I have revised my answer. – 2017-01-13