Mollification of a Lipschitz Convex Function: Smooth Convex Approximations with Bounded Gradients Converging Where the Subgradient is Unique

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, the Euclidean norm $\lVert\,\cdot\,\rVert$, dot product, distance $d_{E}$, topology and closed balls, and the Lebesgue measure $\lambda_{n}$, 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. Derivatives $Df$ of a function of class $C^{1}$ on an open subset of $\mathbb{R}^{n}$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii), and smooth is smoothness on $\mathbb{R}^{n}$.

Let $L$ be a nonnegative real number and let $\phi:\mathbb{R}^{n}\to\mathbb{R}$ be convex on $\mathbb{R}^{n}$ and Lipschitz with constant $L$ from $(\mathbb{R}^{n},d_{E})$ to $\mathbb{R}$ with the absolute-value metric, with subdifferential $\partial_{\mathbb{R}^{n}}\phi$. Let $\delta$ be a positive real number and let $\rho$ be a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$. For a positive real number $\varepsilon$ let $\rho_{\varepsilon}:\mathbb{R}^{n}\to\mathbb{R}$ be given by $\rho_{\varepsilon}(y)=(\varepsilon^{-1})^{n}\rho(\varepsilon^{-1}y)$, a mollifier kernel of radius $\varepsilon\delta$ by Rescaling a Mollifier Kernel; since $\phi$ is continuous on $\mathbb{R}^{n}$ and $\rho_{\varepsilon}$ is continuous and vanishes at every $y$ with $\varepsilon\delta<\lVert y\rVert$, the convolution $\phi_{\varepsilon}=\phi*\rho_{\varepsilon}$ is defined on the whole of $\mathbb{R}^{n}$, the set $\{x\in\mathbb{R}^{n}:\bar{B}(x,\varepsilon\delta)\subseteq\mathbb{R}^{n}\}$ being $\mathbb{R}^{n}$.

1. (Regularity of the mollifications)(i) Let $\varepsilon$ be a positive real number. Then $\phi_{\varepsilon}$ is smooth on $\mathbb{R}^{n}$, convex on $\mathbb{R}^{n}$, and Lipschitz with constant $L$, and

$$\lVert D\phi_{\varepsilon}(x)\rVert\le L\qquad\text{for every }x\in\mathbb{R}^{n}.$$

2. (Convergence of the gradients)(ii) Let $x\in\mathbb{R}^{n}$ and $p\in\mathbb{R}^{n}$ satisfy $\partial_{\mathbb{R}^{n}}\phi(x)=\{p\}$, and let $(\varepsilon_{m})_{m\in\mathbb{N}}$ be a sequence of positive real numbers converging to $0$. Then the sequence $(D\phi_{\varepsilon_{m}}(x))_{m\in\mathbb{N}}$ converges to $p$ in $(\mathbb{R}^{n},d_{E})$.