 Convex positively one-homogeneous functional

This is a proper convex function that has degree-one positive homogeneity, with $J(0)=0$. It need not be even or nonnegative; linear functionals are examples. Its subdifferential satisfiesThe two tests $z=0$ and $z=2u$ prove equality; the subgradient inequality then gives the global upper bound. Conversely that bound and equality prove the subgradient condition. Hence a subgradient at $u$ also supports every nonnegative multiple of $u$. Requiring $J(-u)=J(u)$ gives an absolutely one-homogeneous functional.