${\mathbf{C}}^2$-Lusin approximation of convex functions: one variable case

Abstract

We prove that if $f:(a,b)\to \mathbb{R}$ is convex, then for any $\epsilon >0$ there is a convex function $g\in C^2(a,b)$ such that $|\{f\ne g\}|<\epsilon$ and $\Vert f-g{\Vert }_{\infty }<\epsilon$.