The Area Inequality for the Gradient of a Convex Function

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, used with a natural number $n$ satisfying $1\le n$, let $U\subseteq\mathbb{R}^{n}$ be open and convex, let $f:U\to\mathbb{R}$ be convex on $U$, and let $A\in\mathcal{B}(\mathbb{R}^{n})$ satisfy $A\subseteq U$ and be such that $f$ is twice differentiable at every point of $A$; at $y\in A$ write $Df(y)$ for the first-order coefficient and $D^{2}f(y)$ for the Hessian, and let $\det$ be the determinant. For a Borel $g:\mathbb{R}^{n}\to\mathbb{R}$ with $0\le g$, the integral $\int_{\mathbb{R}^{n}}g\,d\lambda_{n}\in[0,\infty]$ is that of Measure Spaces and the Lebesgue Integral: Standing Notation §integral for the measure space $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$.

1. (Area inequality)¶ Let $h:\mathbb{R}^{n}\to\mathbb{R}$ be Borel with $0\le h(z)$ for every $z\in\mathbb{R}^{n}$, and let $k:\mathbb{R}^{n}\to\mathbb{R}$ be the function with $k(y)=h(Df(y))\det D^{2}f(y)$ for $y\in A$ and $k(y)=0$ for $y\in\mathbb{R}^{n}\setminus A$. Then $k$ is Borel, $0\le k(y)$ for every $y\in\mathbb{R}^{n}$, and