Proof of The Area Inequality for the Gradient of a Convex Function

Each result cited is universally quantified over the data in its own statement. Integrals are with respect to $\lambda_{n}$; for $Z\in\mathcal{B}(\mathbb{R}^{n})$, $\mathbf{1}_{Z}$ is its indicator, Borel by claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions.

Notation from the Borel lemma. For a convex function $F$ on an open convex set $V\subseteq\mathbb{R}^{n}$ and a Borel set $A_{F}\subseteq V$ at every point of which $F$ is twice differentiable, let $\mathcal{G}[F,A_{F}]:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the map with coordinates the functions $g_{i}$ of The Points of Twice Differentiability of a Convex Function: a Borel Set of Full Measure, and Borel Measurability of the Gradient and Hessian on It §borel (for $F$, $V$, $A_{F}$), and $\mathcal{J}[F,A_{F}]$ the function $J$ there; so $\mathcal{G}[F,A_{F}](y)=DF(y)$ and $\mathcal{J}[F,A_{F}](y)=\det D^{2}F(y)$ for $y\in A_{F}$, and both vanish off $A_{F}$. The map $\mathcal{G}[F,A_{F}]$ is Borel (claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets), and $0\le\mathcal{J}[F,A_{F}]$ by The Points of Twice Differentiability of a Convex Function: a Borel Set of Full Measure, and Borel Measurability of the Gradient and Hessian on It §nonnegative. For our $f$ and $A$ write $\mathcal{G}=\mathcal{G}[f,A]$ and $\mathcal{J}=\mathcal{J}[f,A]$. Since $\mathcal{J}$ vanishes off $A$, $k=(h\circ\mathcal{G})\,\mathcal{J}$ on all of $\mathbb{R}^{n}$; it is Borel as a composite and product of Borel functions (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps and claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), and $0\le k$ as $0\le h$ and $0\le\mathcal{J}$.

For the rest of the proof let $F:\mathbb{R}^{n}\to\mathbb{R}$ be convex on $\mathbb{R}^{n}$ and Lipschitz with a constant $\Lambda\ge0$, and let $A_{F}\in\mathcal{B}(\mathbb{R}^{n})$ be a set at every point of which $F$ is twice differentiable; write $\mathcal{G}_{F}=\mathcal{G}[F,A_{F}]$, $\mathcal{J}_{F}=\mathcal{J}[F,A_{F}]$. Steps 1 to 3 prove

first for $u=\mathbf{1}_{W}$ with $W$ open, then for $u=\mathbf{1}_{Z}$ with $Z$ Borel, then in general.

Step 1: open sets. Let $W\subseteq\mathbb{R}^{n}$ be open. Let $\rho$ be a mollifier kernel of radius $1$ (Existence of Mollifier Kernels of Every Radius), let $K$ be the constant of Mollification at a Point of Twice Differentiability: Convergence of the Mollified Gradient and Hessian, and a Hessian Bound for Lipschitz Functions §bound for $n$, $1$ and $\rho$, and for $m\in\mathbb{N}$ let $F_{m}=F*\rho_{1/m}$ as there. By Mollification of a Lipschitz Convex Function: Smooth Convex Approximations with Bounded Gradients Converging Where the Subgradient is Unique §regularity, $F_{m}$ is smooth and convex on $\mathbb{R}^{n}$, so $0_{n}\preceq D^{2}F_{m}(x)$ by A Convex Function of Class $C^2$ has Positive Semidefinite Hessian, and $D^{2}F_{m}(x)\preceq(K\Lambda m)I_{n}$ by Mollification at a Point of Twice Differentiability: Convergence of the Mollified Gradient and Hessian, and a Hessian Bound for Lipschitz Functions §bound. Let $\Phi_{m}(x)=F_{m}(x)+\tfrac{1}{2m}\lVert x\rVert^{2}$. By A Scaled Squared Distance to a Point is of Class $C^2$, with Gradient and Hessian (with $a=0_{\mathbb{R}^{n}}$ and $c=\tfrac{1}{2m}$) and Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $\Phi_{m}$ is of class $C^{2}$ on $\mathbb{R}^{n}$ with $D\Phi_{m}(x)=DF_{m}(x)+m^{-1}x$ and $D^{2}\Phi_{m}(x)=D^{2}F_{m}(x)+m^{-1}I_{n}$, so, by compatibility of $\preceq$ with addition (Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §background),

By The Gradient of a Twice Continuously Differentiable Function with Hessian Pinched between Two Positive Multiples of the Identity is a Bi-Lipschitz Bijection of Euclidean Space with Continuously Differentiable Inverse §bijection and The Gradient of a Twice Continuously Differentiable Function with Hessian Pinched between Two Positive Multiples of the Identity is a Bi-Lipschitz Bijection of Euclidean Space with Continuously Differentiable Inverse §inverse, the gradient map $\nabla\Phi_{m}$ is a bijection of $\mathbb{R}^{n}$ whose components and those of its inverse are of class $C^{1}$, with symmetric positive definite Jacobian matrix $D^{2}\Phi_{m}(x)$ whose inverse matrix is the Jacobian matrix of the inverse at $\nabla\Phi_{m}(x)$. So Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward §integrals applies to $\nabla\Phi_{m}$ and gives, with $u_{m}(y)=\mathbf{1}_{A_{F}}(y)\,\mathbf{1}_{W}(\nabla\Phi_{m}(y))\det D^{2}\Phi_{m}(y)$, which is Borel and satisfies $0\le u_{m}\le\mathbf{1}_{W}(\nabla\Phi_{m})\det D^{2}\Phi_{m}$ (the determinant being positive by Determinants of Positive Definite Matrices: Positivity, the Bound $\log\det A\le\mathrm{tr}\,A-d$, Bounds under Pinching, and the Expansion of $\det(I+tB)$ §positive),

Now fix $y\in A_{F}$ and let $p=\mathcal{G}_{F}(y)$, $H=D^{2}F(y)$. $F$ is continuous (A Lipschitz Map is Uniformly Continuous), so Mollification at a Point of Twice Differentiability: Convergence of the Mollified Gradient and Hessian, and a Hessian Bound for Lipschitz Functions §convergence with $\varepsilon_{m}=m^{-1}$ gives $DF_{m}(y)\to p$ and $D^{2}F_{m}(y)\to H$ in $\mathcal{S}(n)$; since $m^{-1}y\to0_{\mathbb{R}^{n}}$ and $\lVert m^{-1}I_{n}\rVert=m^{-1}\lVert I_{n}\rVert\to0$ (claim 5 of Properties of the Norm of a Symmetric Real Matrix), also $\nabla\Phi_{m}(y)\to p$ and $D^{2}\Phi_{m}(y)\to H$. By Limits and Bounded Sequences of Symmetric Real Matrices §quadratic-form applied with $z=e_{i}$, $e_{j}$, $e_{i}+e_{j}$ and the identity $2X_{ij}=(e_{i}+e_{j})\cdot(X(e_{i}+e_{j}))-X_{ii}-X_{jj}$ for $X\in\mathcal{S}(n)$ (claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum), every entry of $D^{2}\Phi_{m}(y)$ converges to the corresponding entry of $H$, hence $\det D^{2}\Phi_{m}(y)\to\det H=\mathcal{J}_{F}(y)$ by the formula of Determinant of a Real Square Matrix and the limit laws for finite sums and products (Arithmetic of Limits of Real Sequences). If $p\in W$, then, $W$ being open, $\nabla\Phi_{m}(y)\in W$ for all large $m$, so $u_{m}(y)=\det D^{2}\Phi_{m}(y)$ for all large $m$ and $u_{m}(y)\to\mathcal{J}_{F}(y)=\mathbf{1}_{A_{F}}(y)\mathbf{1}_{W}(p)\mathcal{J}_{F}(y)$. If $p\notin W$, then $\mathbf{1}_{A_{F}}(y)\mathbf{1}_{W}(p)\mathcal{J}_{F}(y)=0\le u_{m}(y)$ for all $m$; and for $y\notin A_{F}$ both sides vanish. So in every case the $\liminf$ of Fatou's Lemma satisfies $\liminf_{m}u_{m}(y)\ge\mathbf{1}_{A_{F}}(y)\mathbf{1}_{W}(\mathcal{G}_{F}(y))\mathcal{J}_{F}(y)$, and Fatou's lemma together with (1) and monotonicity of the integral gives

Step 2: Borel sets. Let $Z\in\mathcal{B}(\mathbb{R}^{n})$; we may suppose $\lambda_{n}(Z)<\infty$. For $R\in\mathbb{N}$ let $V_{R}$ be the open ball $B(0_{\mathbb{R}^{n}},R)$, which lies in the closed ball of radius $R$ and so has finite measure (The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$ §borel). The measure $\lambda^{R}$ with density $\mathbf{1}_{V_{R}}$ with respect to $\lambda_{n}$ (claim 3 of Image Measures, Measures with Densities, and Change of Variables) satisfies $\lambda^{R}(C)=\lambda_{n}(C\cap V_{R})$ and is a finite Borel measure on $(\mathbb{R}^{n},d_{E})$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces). Let $\eta>0$. By Inner and Outer Regularity of a Finite Borel Measure on a Metric Space, and Lipschitz Approximation of Indicators §outer there is an open $O\supseteq Z\cap V_{R}$ with $\lambda^{R}(O)\le\lambda^{R}(Z\cap V_{R})+\eta$ (Approximation Property of the Supremum and the Infimum in $\mathbb{R}$, claim 3). The set $W=O\cap V_{R}$ is open, contains $Z\cap V_{R}$, and $\lambda_{n}(W)=\lambda^{R}(O)\le\lambda_{n}(Z)+\eta$. By monotonicity of the integral and Step 1,

As $\eta>0$ is arbitrary, the left side is at most $\lambda_{n}(Z)$. As $R\to\infty$, $\mathbf{1}_{Z\cap V_{R}}$ increases pointwise to $\mathbf{1}_{Z}$ (every point lies in some $V_{R}$, by The Archimedean Property of the Real Numbers), so the monotone convergence theorem gives $(\ast)$ for $u=\mathbf{1}_{Z}$.

Step 3: nonnegative Borel functions. Let $u$ be Borel with $0\le u$. By Approximation of Measurable Functions by Simple Functions §nonnegative there are nonnegative simple Borel functions $s_{j}=\sum_{l}c_{jl}\mathbf{1}_{Z_{jl}}$ (finite sums, $c_{jl}\ge0$, $Z_{jl}\in\mathcal{B}(\mathbb{R}^{n})$) increasing pointwise to $u$. By additivity and homogeneity of the integral of nonnegative functions (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) and Step 2, $\int\mathbf{1}_{A_{F}}(s_{j}\circ\mathcal{G}_{F})\mathcal{J}_{F}=\sum_{l}c_{jl}\int\mathbf{1}_{A_{F}}(\mathbf{1}_{Z_{jl}}\circ\mathcal{G}_{F})\mathcal{J}_{F}\le\sum_{l}c_{jl}\lambda_{n}(Z_{jl})=\int s_{j}$. Since $\mathbf{1}_{A_{F}}(s_{j}\circ\mathcal{G}_{F})\mathcal{J}_{F}$ increases pointwise to $\mathbf{1}_{A_{F}}(u\circ\mathcal{G}_{F})\mathcal{J}_{F}$, monotone convergence on both sides gives $(\ast)$.

Step 4: the general case. If $A$ is empty then $k=0$ and there is nothing to prove; otherwise $U$ is nonempty, and we fix $x_{0}\in U$ and $q_{0}\in\partial_{U}f(x_{0})$ (The Subdifferential of a Convex Function on an Open Convex Set is Nonempty §nonempty). Let $L\in\mathbb{N}$ with $\lVert q_{0}\rVert<L$, let $f^{L}$ be the Lipschitz truncation of $f$ at level $L$, convex on $\mathbb{R}^{n}$ and Lipschitz with constant $L$ by The Lipschitz Truncation of a Convex Function: a Global Lipschitz Convex Minorant Agreeing with It Where the Slope is Small §minorant, and let $A_{L}=\{y\in A:\lVert\mathcal{G}(y)\rVert<L\}$, a Borel set ($\mathcal{G}$ and the continuous norm being Borel). For $y\in A_{L}$, Subgradients near a Point of Twice Differentiability of a Convex Function, and Invariance of the Second-Order Expansion under Lipschitz Truncation §truncation shows that $f^{L}$ is twice differentiable at $y$ with first-order coefficient $Df(y)$ and Hessian $D^{2}f(y)$; by the uniqueness recorded in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §twice-differentiable, $\mathcal{G}[f^{L},A_{L}]=\mathcal{G}$ and $\mathcal{J}[f^{L},A_{L}]=\mathcal{J}$ on $A_{L}$. Applying $(\ast)$ to $F=f^{L}$, $A_{F}=A_{L}$ and $u=h$ gives

As $L\to\infty$, $\mathbf{1}_{A_{L}}k$ increases pointwise to $\mathbf{1}_{A}k=k$ (every $y\in A$ has $\lVert\mathcal{G}(y)\rVert<L$ for large $L$, by The Archimedean Property of the Real Numbers), and monotone convergence gives $\int k\,d\lambda_{n}\le\int h\,d\lambda_{n}$.