Intuition explanation of the gauge function in optimization

Question

The gauge function is defined as $\gamma(x|C) = \inf\left\{\lambda \ge 0 | x \in \lambda C \right\}$ where $C$ is a convex set. I read that it is a generalization of norm concept but I can't understand that. Is there an intuitive explanation of what this function does in optimization?

Answer 1

First, usually there is a condition that $0 \in \mathring C$ or something similar ; else there are some points who will have infinite gauge value.

The relation to the norm is this : take $C = B_N(0,1)$ the ball for the norm $N$, then $N(x) = \gamma(x|C)$. Indeed, it's the smallest value $\lambda$ such that $x \in B_N(0,\lambda)$.

Actually, if the convex set $C$ is bounded and symmetric ($x \in C \Leftrightarrow -x \in C)$, then $\gamma(\cdot,C)$ is a norm (without the bounded attribute, it's only a seminorm). So the concepts are very similar. You can simply see a gauge function as an asymmetric seminorm.