Euler identity for a convex one-homogeneous functional
= Euler identity for a convex one-homogeneous functional
{c}
For an <absolutely one-homogeneous functional> $J$ that is convex and $p\in\partial J(u)$,
$$
J(u)=\langle p,u\rangle.
$$
The <subgradient inequality> at zero gives $\langle p,u\rangle\geq J(u)$, and at $2u$ gives the reverse inequality. This is an Euler identity that needs no differentiability.