Lusin-type properties of convex functions and convex bodies

TL;DR

This paper demonstrates that convex functions and convex bodies can be approximated arbitrarily closely by smoother convex functions and bodies of class C^{1,1}, with applications to convex hypersurfaces and geometric measure theory.

Contribution

It establishes new approximation results for convex functions and convex bodies by C^{1,1} smooth counterparts, characterizing when such approximations are possible.

Findings

Convex functions can be approximated on finite measure sets by C^{1,1} convex functions.

Convex bodies can be approximated in measure by C^{1,1} convex bodies.

Characterization of convex functions that can be approximated by locally C^{1,1} functions.

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 $\varepsilon>0$ there is a convex function $g:\mathbb{R}^n\to\mathbb{R}$ of class $C^{1,1}$ such that $\mathcal{L}^n(\{x\in A:\, f(x)\neq g(x)\})<\varepsilon$. 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 $f:\mathbb{R}^n\to\mathbb{R}$ is a convex function and $f$ is not of class $C^{1,1}_{\rm loc}$, then for any $\varepsilon>0$ there is a convex function $g:\mathbb{R}^n\to\mathbb{R}$ of class $C^{1,1}_{\rm loc}$ such that $\mathcal{L}^n(\{x\in \mathbb{R}^n:\, f(x)\neq g(x)\})<\varepsilon$ if and only if $f$ is essentially coercive, meaning that $\lim_{|x|\to\infty}f(x)-\ell(x)=\infty$ 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 $\varepsilon>0$ there exists a convex hypersurface $S_{\varepsilon}$ of class $C^{1,1}_{\textrm{loc}}$ such that $\mathcal{H}^{n-1}(S\setminus S_{\varepsilon})<\varepsilon$.