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