The Lipschitz Truncation of a Convex Function: a Global Lipschitz Convex Minorant Agreeing with It Where the Slope is Small

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers and sequences, and the Euclidean norm $\lVert\,\cdot\,\rVert$, dot product, distance $d_{E}$ and topology, are as fixed there. The space $\mathbb{R}^{n}$ is open in itself by claim 1 of Euclidean Space Is Open in Itself and is convex, every convex combination of two of its points being again one of its points.

Let $G\subseteq\mathbb{R}^{n}$ be open and convex and nonempty, let $\phi:G\to\mathbb{R}$ be convex on $G$, with subdifferential $\partial_{G}\phi$, and let $L$ be a real number for which there are $x_{0}\in G$ and $q_{0}\in\partial_{G}\phi(x_{0})$ with $\lVert q_{0}\rVert\le L$; such an $L$ exists, since $\partial_{G}\phi(x_{0})$ is nonempty for every $x_{0}\in G$ by The Subdifferential of a Convex Function on an Open Convex Set is Nonempty. The Lipschitz truncation of $\phi$ at level $L$ is the function $\phi^{L}:\mathbb{R}^{n}\to\mathbb{R}$ given by

$$\phi^{L}(x)=\inf\bigl\{\phi(z)+L\lVert x-z\rVert\ :\ z\in G\bigr\}\qquad(x\in\mathbb{R}^{n}),$$

the infimum existing in $\mathbb{R}$ by claim 1 below.

1. (A Lipschitz convex minorant)(i) For every $x\in\mathbb{R}^{n}$ the set $\{\phi(z)+L\lVert x-z\rVert:z\in G\}$ is nonempty and bounded below, so that $\phi^{L}(x)$ is a real number. The function $\phi^{L}$ is convex on $\mathbb{R}^{n}$ and is Lipschitz with constant $L$ from $(\mathbb{R}^{n},d_{E})$ to $\mathbb{R}$ with the absolute-value metric, and $\phi^{L}(x)\le\phi(x)$ for every $x\in G$.

2. (Agreement and subgradients where the slope is small)(ii) Let $x\in G$ and let $p\in\partial_{G}\phi(x)$ satisfy $\lVert p\rVert\le L$. Then

$$\phi^{L}(x)=\phi(x),\qquad p\in\partial_{\mathbb{R}^{n}}\phi^{L}(x).$$

3. (A unique subgradient is inherited)(iii) Let $x\in G$ and suppose that $\partial_{G}\phi(x)=\{p\}$ with $\lVert p\rVert\le L$. Then $\partial_{\mathbb{R}^{n}}\phi^{L}(x)=\{p\}$.