Lusin-Type Properties of Convex Functions and Convex Bodies

Abstract

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 ${\mathcal {H}}^{n-1}(S\setminus S_{\varepsilon })<\varepsilon$.