Absolutely one-homogeneous functional
= Absolutely one-homogeneous functional
A functional $J$ is absolutely one-homogeneous when $J(cu)=|c|J(u)$ for every scalar $c$. For a convex such functional,
$$
p\in\partial J(u)
\quad\Longleftrightarrow\quad
\langle p,u\rangle=J(u)
\text{ and }
\langle p,v\rangle\leq J(v)\text{ for every }v.
$$