Gauge function

A nonnegative, convex function, from $LaTeX: \mathbb{R}^n$ to the extended reals that is positively homogeneous and $LaTeX: f(0)=0$. A norm is a gauge function. Here is an example that is not a norm:
$LaTeX: f(x,y,z) = \left\{ \begin{array}{rl}
 \| (x-z,\; x-y) \|_2, & 2x - y - z = 0 \\ \\ \infty, & \mbox{ otherwise.} \end{array} \right.