When the function $f$ is differentiable or piecewise affine, the measurable function $f^{\vee}\circ\partial f$ can be made explicit.

1. (1)

   Let $f\in{\mathcal{C}}^{2}(N_{\mathbb{R}})$. Proposition 3.94 and the change of variables formula imply $g\circ\partial f=g\circ\nabla f$. For the particular case when $g=f^{\vee}$, Theorem 3.52(4) implies, for $u\in N_{\mathbb{R}}$,

   $$f^{\vee}\circ\partial f(u)=\langle\nabla f(u),u\rangle-f(u).$$

2. (2)

   Let $f$ a piecewise affine concave function on $N_{\mathbb{R}}$. By Proposition 3.95, ${\mathcal{M}}_{M}(f)$ is supported in the finite set $\Pi(f)^{0}$ and so is ${\mathcal{M}}_{M,g}(f)$. For $v\in\Pi(f)^{0}$ write $v^{*}\in\Pi(f^{\vee})^{n}$ for the dual polyhedron. Then $g\circ\partial f(v)=\frac{1}{\operatorname{vol}_{M}(v^{*})}\int_{v^{*}}g\,\text{\rm d}\operatorname{vol}_{M}$, which implies

   $$f^{\vee}\circ\partial f(v)=\frac{1}{\operatorname{vol}_{M}(v^{*})}\int_{v^{*}}\langle x,v\rangle\,\text{\rm d}\operatorname{vol}_{M}-f(v).$$

   The function $f^{\vee}\circ\partial f$ is defined as a ${\mathcal{M}}_{M}(f)$-measurable function. Therefore, only its values at the points $v\in\Pi(f)^{0}$ are well defined. Nevertheless, we can extend the function $f^{\vee}\circ\partial f$ to the whole $N_{\mathbb{R}}$ by writing

   $$f^{\vee}\circ\partial f(u)=\frac{1}{\operatorname{vol}_{\mu}(\partial f(u))}\int_{\partial f(u)}\langle x,u\rangle\,\text{\rm d}\mu-f(u)$$

   for any Haar measure $\mu$ on the affine space determined by $\partial f(u)$.