Lusin-type properties of convex functions and convex bodies

We prove that if $f:{\mathbb{R}}^n\to \mathbb{R}$ is convex and $A\subset {\mathbb{R}}^n$ has finite measure, then for any $\epsilon >0$ there is a convex function $g:{\mathbb{R}}^n\to \mathbb{R}$ of class $C^{1,1}$ such that ${\mathcal{L}}^n(\{x\in A:f(x)\mathrm{\ne }g(x)\})<\epsilon$. As an application we deduce that if $W\subset {\mathbb{R}}^n$ is a compact convex body then, for every $\epsilon >0$, there exists a convex body $W_{\epsilon }$ of class $C^{1,1}$ such that $$\mathcal{H}$^{n-1}\left(\partial W\setminus \partial W_{\varepsilon}\right)< \varepsilon$\mathrm{.}Wealsoshowthatif$f:$\mathbb{R}$^n\to$\mathbb{R}$$isaconvexfunctionand$f$isnotofclass$C^{1,1}{\rm loc}$,thenforany$\varepsilon>0$thereisaconvexfunction$g:$\mathbb{R}$^n\to$\mathbb{R}$$ofclass$C^{1,1}{\rm loc}$suchthat$$\mathcal{L}$^n({x\in $\mathbb{R}$^n:, f(x)\neq g(x)})<\varepsilon$ if and only if $f$ is essentially coercive, meaning that ${\lim }_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}f(x)-\mathrm{\ell }(x)=\mathrm{\infty }$ for some linear function $\mathrm{\ell }$. A consequence of this result is that, if $S$ is the boundary of some convex set with nonempty interior (not necessarily bounded) in ${\mathbb{R}}^n$ and $S$ does not contain any line, then for every $\epsilon >0$ there exists a convex hypersurface $S_{\epsilon }$ of class $C_{\text{loc}}^{1,1}$ such that ${\mathcal{H}}^{n-1}(S\setminus S_{\epsilon })<\epsilon$.