Hessians of Two Convex Functions with Mutually Inverse Subgradients are Inverse Matrices at a Density Point

In the setting of Euclidean Space and Lebesgue Measure: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, used with a natural number $n$ satisfying $1\le n$, let density point have the meaning fixed there, let $\det$ be the determinant, and let positive definiteness and the inverse matrix be as defined there. Let $U,U'\subseteq\mathbb{R}^{n}$ be open and convex, let $f:U\to\mathbb{R}$ be convex on $U$ and $g:U'\to\mathbb{R}$ convex on $U'$, with subdifferentials $\partial_{U}f$ and $\partial_{U'}g$. Let $x\in U$, $y\in U'$ and $B,B'\in\mathcal{S}(n)$ be such that $f$ is twice differentiable at $x$ with first-order coefficient $y$ and Hessian $B$, and $g$ is twice differentiable at $y$ with first-order coefficient $x$ and Hessian $B'$. Let $E\subseteq U$ be a set such that for every $x'\in E$ there is $q\in U'$ with

and suppose that $x$ is a density point of $E$.

1. (Inverse Hessians)¶ $B'B=I_{n}$ and $BB'=I_{n}$, so that each of $B$ and $B'$ is invertible with inverse matrix the other; both are positive definite, and $\det B\cdot\det B'=1$.