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

TL;DR

This paper demonstrates that convex functions on an interval can be closely approximated by twice continuously differentiable convex functions, with the approximation arbitrarily precise in both measure and uniform norm.

Contribution

It establishes a $C^2$-Lusin approximation result for convex functions in one variable, extending the understanding of smooth approximation in convex analysis.

Findings

Convex functions can be approximated by $C^2$ convex functions within any desired accuracy.

The approximation preserves convexity and can be made arbitrarily close in measure and uniform norm.

This result provides a new tool for smooth approximation in convex analysis and optimization.

Abstract

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