Proposition 21.11.34.8B18F2B2DE3B Let $F\in \mathrm{Conv}^{\mathrm{prop}}(N)$. Then
\[ \nabla F(N)\subseteq \operatorname{Dom}F^*. \]
If moreover $F$ is closed, we have
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21.11.35
\begin{equation} \operatorname{RelInt}\operatorname{Dom}F^*\subseteq \nabla F(N). \end{equation}
In particular, if $F$ is a proper closed convex function on $N$, then
\[ \overline{\nabla F(N)}=\overline{\operatorname{Dom}F^*}. \]
Proof. If $p\in\partial F(x)$, the subgradient inequality gives $F^*(p)=\langle p,x\rangle-F(x)<+\infty$, proving the first inclusion. Now assume $F$ is closed and let $p$ lie in the relative interior of $\operatorname{Dom}F^*$. The conjugate is closed and proper by Fenchel duality, so it has a subgradient $x$ at $p$. The same duality theorem gives $p\in\partial F(x)$. Thus the relative interior is contained in the subgradient image. Taking closures in the two inclusions and using that a nonempty convex set has the same closure as its relative interior proves the equality of closures.
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