We prove that if \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is convex and \(A\subset {\mathbb {R}}^n\) has finite measure, then for any \(\varepsilon >0\) there is a convex function \(g:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) of class \(C^{1,1}\) such that \({\mathcal {L}}^n(\{x\in A:\, f(x)\ne g(x)\})<\varepsilon \). As an application we deduce that if \(W\subset {\mathbb {R}}^n\) is a compact convex body then, for every \(\varepsilon >0\), there exists a convex body \(W_{\varepsilon }\) of class \(C^{1,1}\) such that \({\mathcal {H}}^{n-1}\left( \partial W\setminus \partial W_{\varepsilon }\right) < \varepsilon \). We also show that if \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is a convex function and f is not of class \(C^{1,1}_{\mathrm{loc}}\), then for any \(\varepsilon >0\) there is a convex function \(g:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) of class \(C^{1,1}_{\mathrm{loc}}\) such that \({\mathcal {L}}^n(\{x\in {\mathbb {R}}^n:\, f(x)\ne g(x)\})<\varepsilon \) if and only if f is essentially coercive, meaning that \(\lim _{|x|\rightarrow \infty }f(x)-\ell (x)=\infty \) for some linear function \(\ell \). 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 \(\varepsilon >0\) there exists a convex hypersurface \(S_{\varepsilon }\) of class \(C^{1,1}_{\text {loc}}\) such that \({\mathcal {H}}^{n-1}(S\setminus S_{\varepsilon })<\varepsilon \).

我们证明如果\（f：{\ mathbb {R}} ^ n \ rightarrow {\ mathbb {R}} \）是凸的并且\（A \ subset {\ mathbb {R}} ^ n \）具有有限度量，那么对于任意\（\ varepsilon> 0 \）有一个凸函数\（克：{\ mathbb {R}} ^ N \ RIGHTARROW {\ mathbb {R}} \）类的\（C ^ {1， 1} \）这样\（{\ mathcal {L}} ^ n（\ {x \ in A：\，f（x）\ ne g（x）\}）<\ varepsilon \）。作为一个应用，我们推论如果\（W \ subset {\ mathbb {R}} ^ n \）是一个紧凑的凸体，那么对于每个\（\ varepsilon> 0 \），都会存在一个凸体\（W_ { \ （C ^ {1,1} \）类的\ varepsilon} \）使得\（{\ mathcal {H}} ^ {n-1} \ left（\ partial W \ setminus \ partial W _ {\ varepsilon} \ right）<\ varepsilon \）。我们还表明，如果\（f：{\ mathbb {R}} ^ n \ rightarrow {\ mathbb {R}} \）是凸函数，而f不是\（C ^ {1,1} _ { \ mathrm {LOC}} \），那么对于任何\（\ varepsilon> 0 \）有一个凸函数\（克：{\ mathbb {R}} ^ N \ RIGHTARROW {\ mathbb {R}} \）的类\（C ^ {1,1} _ {\ mathrm {loc}} \）使得\（{\ mathcal {L}} ^ n（\ {x \ in {\ mathbb {R}} ^ n：\ ，f（x）\ ne g（x）\}）<\ varepsilon \）当且仅当f本质上是强制性的，这意味着\（\ lim _ {| x | \ rightarrow \ infty} f（x）-\ ell（x）= \ infty \）对于一些线性函数\（\ ell \）。该结果的结果是，如果S是\（{\ mathbb {R}} ^ n \）中具有非空内部（不一定有界）的某个凸集的边界，并且S不包含任何线，则对于每个\（\ varepsilon> 0 \）存在类\（C ^ {1,1} _ {\ text {loc}} \\）的凸超曲面\（S _ {\ varepsilon} \），使得\（{\ mathcal {H}} ^ {n-1}（S \ setminus S _ {\ varepsilon}）<\ varepsilon \）。