Theorem 21.11.33A57F525B7ABC

Proof. Subgradients at points outside the effective domain are empty by definition. On the relative interior, existence and the directional formula follow from local convex regularity applied in the affine span $A$ of the domain. To view a subgradient as a functional on $N$, extend its restriction to the direction space of $A$ by a linear functional. Any functional vanishing on that direction space may be added without changing the subgradient inequality. Thus for a direction outside that space the supremum in the directional formula is $+\infty$, just as the one-sided directional derivative is; for tangent directions it is the relative formula.

At an interior point of the domain, local Lipschitz continuity bounds all subgradients: if $|F(x+v)-F(x)|\leq L|v|$ for small $v$, the subgradient inequality for $v$ and $-v$ gives $|\langle p,v\rangle|\leq L|v|$, hence $|p|\leq L$. Conversely, suppose the subgradient set is nonempty at a point $x$ of the domain which is not interior. A supporting hyperplane to the closure of the convex domain at $x$ supplies a nonzero $q$ with $\langle q,y-x\rangle\leq0$ for every $y$ in the domain. If $p$ is a subgradient, then so is $p+tq$ for every $t\geq0$, so the set is unbounded. Supporting hyperplanes also exist when the domain lies in a proper affine subspace, by taking an annihilating functional. This proves the final equivalence.