= Convex positively one-homogeneous functional
{title2=$J(tu)=tJ(u)\quad(t\geq0)$}
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> satisfies
$$
p\in\partial J(u)\iff \langle p,u\rangle=J(u)\ \text{and}\ \langle p,z\rangle\leq J(z)\ \text{for every }z.
$$
The 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>.
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