  General $(\alpha, \beta)$ norms are an important class of Minkowski norms
which contains the original $(\alpha, \beta)$ norms. In this note, by studying
the behavior of the Darboux curves of the indicatrix, we give a
characterization of 3-dimensional general $(\alpha, \beta)$ norms. By studying
the isoperimetric properties of the indicatrix, as well as the isoperimetric
inequalities in a Minkowski space, we give some global geometric quantities
which characterizes Randers norms of arbitrary dimensions.
