GAUGE FUNCTION

Gauge function - A nonnegative, convex function , from R^n to the extended reals, that is positively homogeneous and f(0)=0. A norm is a gauge function. Here is an example that is not a norm:

 _ | ||(x-z), (x-y)||_2 if 2x - y - z = 0; f(x, y, z) = | |_ infinity otherwise. 

Here is a graphic .