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

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. For a positive real $\varepsilon$ and $x\in\mathbb{R}^{n}$ write $h_{x}(w)=\phi(x-w)\rho_{\varepsilon}(w)$, so that $\phi_{\varepsilon}(x)=\int_{\mathbb{R}^{n}}h_{x}\,d\lambda_{n}$ by Convolution of a Continuous Function with a Compactly Supported Continuous Kernel. By Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$ the kernel $\rho_{\varepsilon}$ is nonnegative with $\int_{\mathbb{R}^{n}}\rho_{\varepsilon}\,d\lambda_{n}=1$. Write $\partial\phi=\partial_{\mathbb{R}^{n}}\phi$ and $\partial\phi_{\varepsilon}=\partial_{\mathbb{R}^{n}}\phi_{\varepsilon}$.

Step 1 (Claim 1). Let $\varepsilon$ be a positive real number.

Smoothness. The kernel $\rho_{\varepsilon}$ is smooth, so $\phi_{\varepsilon}$ is smooth on $\mathbb{R}^{n}$ by Convolution with a $C^k$ Kernel is of Class $C^k$, and in particular of class $C^{1}$, hence differentiable at every point by Continuous Partial Derivatives Imply Differentiability, with gradient $D\phi_{\varepsilon}(x)$ as fixed in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii).

Convexity. By Semiconvex Function on a Convex Subset of $\mathbb{R}^n$ a function is convex on a convex set exactly when it is semiconvex there with constant $0$, since adding $\tfrac{0}{2}\lVert x\rVert^{2}$ changes nothing. So $\phi$ is semiconvex on the open convex set $\mathbb{R}^{n}$ with constant $0$, and Mollification Preserves Semiconvexity, applied with $\Omega=\mathbb{R}^{n}$, the kernel $\rho_{\varepsilon}$ of radius $\varepsilon\delta$ and the constant $0$, gives that $\phi_{\varepsilon}$ is semiconvex with constant $0$ on $\{x:\bar{B}(x,\varepsilon\delta)\subseteq\mathbb{R}^{n}\}=\mathbb{R}^{n}$, that is, convex on $\mathbb{R}^{n}$.

Lipschitz constant. Let $x,y\in\mathbb{R}^{n}$. For every $w$ one has $|h_{x}(w)-h_{y}(w)|=|\phi(x-w)-\phi(y-w)|\,\rho_{\varepsilon}(w)\le L\,d_{E}(x,y)\,\rho_{\varepsilon}(w)$, using $\rho_{\varepsilon}\ge0$, claim 4 of Properties of the Absolute Value in an Ordered Field, the Lipschitz hypothesis on $\phi$, and $d_{E}(x-w,y-w)=d_{E}(x,y)$, which holds by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Hence, by claims 1 and 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and the unit mass of $\rho_{\varepsilon}$,

$$|\phi_{\varepsilon}(x)-\phi_{\varepsilon}(y)|=\Bigl|\int_{\mathbb{R}^{n}}(h_{x}-h_{y})\,d\lambda_{n}\Bigr|\le L\,d_{E}(x,y)\int_{\mathbb{R}^{n}}\rho_{\varepsilon}\,d\lambda_{n}=L\,d_{E}(x,y),$$

so $\phi_{\varepsilon}$ is Lipschitz with constant $L$.

Bound on the gradient. Fix $x\in\mathbb{R}^{n}$. Since $\phi_{\varepsilon}$ is convex on the open convex set $\mathbb{R}^{n}$ and differentiable at $x$, claim 4 of Elementary Calculus of the Subdifferential of a Convex Function gives $\partial\phi_{\varepsilon}(x)=\{D\phi_{\varepsilon}(x)\}$. Applying claim 2 of that lemma with $y_{0}=x$, $r=1$ and $M=L$, which is legitimate because $\bar{B}(x,2)\subseteq\mathbb{R}^{n}$ and $\phi_{\varepsilon}$ is Lipschitz with constant $L$ on all of $\mathbb{R}^{n}$, gives $\lVert q\rVert\le L$ for every $q\in\partial\phi_{\varepsilon}(x)$. Hence $\lVert D\phi_{\varepsilon}(x)\rVert\le L$.

Step 2 (Claim 2). Let $x$, $p$ and $(\varepsilon_{m})_{m\in\mathbb{N}}$ be as in claim 2 and put $q_{m}=D\phi_{\varepsilon_{m}}(x)$, so $\lVert q_{m}\rVert\le L$ for every $m$ by claim 1.

Suppose that $(q_{m})_{m}$ does not converge to $p$. Then there are a positive real $\varepsilon$ and a subsequence $(q_{m_{i}})_{i}$ with $\varepsilon\le d_{E}(q_{m_{i}},p)$ for every $i$. That subsequence is bounded, so The Bolzano-Weierstrass Theorem in $\mathbb{R}^n$ gives a further subsequence $(q_{m_{i_{l}}})_{l}$ converging to some $q\in\mathbb{R}^{n}$, and $\varepsilon\le d_{E}(q,p)$ by Order Properties of Limits of Real Sequences applied to the sequence of distances, which converges to $d_{E}(q,p)$ by Continuity of the Projections and of the Distance Function on a Product Metric Space; in particular $q\ne p$.

Let $y\in\mathbb{R}^{n}$. Since $\phi_{\varepsilon_{m}}$ is convex and $q_{m}\in\partial\phi_{\varepsilon_{m}}(x)$ by Step 1,

$$\phi_{\varepsilon_{m}}(y)\ge\phi_{\varepsilon_{m}}(x)+q_{m}\cdot(y-x)\qquad\text{for every }m\in\mathbb{N}. \tag{$*$}$$

The set $K=\bar{B}(0_{\mathbb{R}^{n}},r)$ with $r=\max(\lVert x\rVert,\lVert y\rVert)$ is compact by The Heine-Borel Theorem in $\mathbb{R}^n$, and Compactness of Closed Balls and the Unit Sphere and contains $x$ and $y$ by claim 1 of Elementary Properties of the Maximum of Two Elements and claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Let $\eta$ be a positive real number. By claim 2 of Mollification Converges Uniformly on Compact Subsets, applied to $\phi$ on $\Omega=\mathbb{R}^{n}$ with the kernel $\rho$, the compact set $K$ and $\eta$, there is a positive real $\varepsilon_{0}$ such that $|\phi_{\varepsilon'}(u)-\phi(u)|<\eta$ for every $u\in K$ and every positive real $\varepsilon'<\varepsilon_{0}$; since $(\varepsilon_{m})_{m}$ converges to $0$, this holds for all large $m$. Hence $(\phi_{\varepsilon_{m}}(y))_{m}$ converges to $\phi(y)$ and $(\phi_{\varepsilon_{m}}(x))_{m}$ converges to $\phi(x)$.

The sequence $\bigl(q_{m_{i_{l}}}\cdot(y-x)\bigr)_{l}$ converges to $q\cdot(y-x)$, by The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite together with claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square and claims 1 and 3 of Arithmetic of Limits of Real Sequences, the map $u\mapsto u\cdot(y-x)$ being a finite sum of scalar multiples of the coordinates. Along the subsequence $(m_{i_{l}})_{l}$ the sequences $(\phi_{\varepsilon_{m}}(y))_{m}$ and $(\phi_{\varepsilon_{m}}(x))_{m}$ still converge to $\phi(y)$ and $\phi(x)$, by A Subsequence of a Convergent Sequence Has the Same Limit. Passing to the limit in $(*)$ along that subsequence and using Order Properties of Limits of Real Sequences together with claim 1 of Arithmetic of Limits of Real Sequences,

$$\phi(y)\ge\phi(x)+q\cdot(y-x).$$

As $y\in\mathbb{R}^{n}$ was arbitrary, $q\in\partial\phi(x)=\{p\}$ by Subdifferential of a Real-Valued Function on a Convex Subset of $\mathbb{R}^n$ (i), so $q=p$, contradicting $q\ne p$. Hence $(D\phi_{\varepsilon_{m}}(x))_{m}$ converges to $p$.