Lusin-type properties of convex functions and convex bodies

Abstract

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 ${\mathcal{L}}^n(\{x\in A:f(x)\ne g(x)\})<\epsilon$. 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 ${\mathcal{H}}^{n-1}\left(\partial W\setminus \partial W_{\epsilon }\right)<\epsilon$. We also show that if $f:{\mathbb{R}}^n\to \mathbb{R}$ is a convex function and $f$ is not of class $C_{\mathrm{loc}}^{1,1}$, then for any $\epsilon >0$ there is a convex function $g:{\mathbb{R}}^n\to \mathbb{R}$ of class $C_{\mathrm{loc}}^{1,1}$ such that ${\mathcal{L}}^n(\{x\in {\mathbb{R}}^n:f(x)\ne g(x)\})<\epsilon$ if and only if $f$ is essentially coercive, meaning that ${\lim }_{|x|\to \infty }f(x)-\ell (x)=\infty$ 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 $\epsilon >0$ there exists a convex hypersurface $S_{\epsilon }$ of class $C_{\text{loc}}^{1,1}$ such that ${\mathcal{H}}^{n-1}(S\setminus S_{\epsilon })<\epsilon$.

15 pages