Proposition 21.11.5284CD4F8DE81D

Proof. The measure normalization here is $n!$ times the Lebesgue measure of the subgradient image. The density formula for locally $C^{1,1}$ convex functions proves that this measure is $n!\det D^2F\,\mathrm dx$. Its hypotheses hold because $C^{1,1}$ implies that the gradient is locally Lipschitz, and the lattice coordinates identify $\mathrm dx$ with the normalized $\mathrm d\operatorname{vol}$. The Hessian is its almost-everywhere derivative, equivalently the density of the distributional Hessian. Thus the formula includes nonsmooth $C^{1,1}$ functions and has the required factor $n!$.