$\mathbf{C^{2}}$ -Lusin approximation of strongly convex functions

Abstract

We prove that if \(u:\mathbb{R}^{n}\to \mathbb{R}\) is strongly convex, then for every \(\varepsilon >0\) there is a strongly convex function \(v\in C^{2}(\mathbb{R}^{n})\) such that \(|\{u\neq v\}|<\varepsilon \) and \(\Vert u-v\Vert _{\infty}<\varepsilon \).