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

TL;DR

The paper proves that any convex function $f:(a,b)\to\mathbb{R}$ on a one-dimensional interval can be uniformly approximated by a convex function $g\in C^2(a,b)$ in the Lusin sense: for every $\varepsilon_o>0$ and continuous $\varepsilon:(a,b)\to(0,\infty)$ there exists such a $g$ with $|\{f\neq g\}|<\varepsilon_o$ and $|f-g|<\varepsilon$. The method combines a Whitney $C^2$ approximation with a convex patching procedure on the complement of a large set, and a careful analysis of endpoint and interior interval corrections to preserve convexity and achieve $C^2$ regularity. The approach first treats a special case via endpoint interior corrections and then extends to the general case by a gluing argument over subintervals. This removes the need for local strong convexity in 1D and raises the question of higher-dimensional analogues, which the authors partly address through a conjecture and open problems.

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 $\Vert f-g\Vert_\infty<\varepsilon$.

Figures (2)

• Figure 1: The graph of $h_i$ when $\tau_i\in (0,c_i/2]$ (left) and $\tau_i\in [c_i/2, c_i)$ (right).
• Figure 2: Lemma \ref{['T10']}: the graph of $g$ must be contained in the shaded area.

Theorems & Definitions (17)

• Theorem 1.1: ACHAH
• Theorem 1.2: ADH
• Theorem 1.3
• Conjecture 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof