Alexandrov's Theorem: a Convex Function on $\mathbb{R}^n$ is Twice Differentiable Almost Everywhere

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, the Euclidean norm $\lVert\,\cdot\,\rVert$, dot product, distance $d_{E}$ and notion of openness, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, Lebesgue measure $\lambda_{n}$, the notion of a null subset, and the conventions on images $T(A)$ and on Lipschitz maps are as fixed there. Write $\mathcal{S}(n)$ for the set of symmetric real $n\times n$ matrices, $0_{n}$ for the real $n\times n$ matrix all of whose entries are $0$, which belongs to $\mathcal{S}(n)$, and $\preceq$ for the positive semidefinite ordering on $\mathcal{S}(n)$.

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be convex on $\mathbb{R}^{n}$, which is an open convex subset of itself. Twice differentiability at a point and the Hessian $D^{2}f(y)$ are as fixed there. Then the following hold.

1. (Twice differentiability almost everywhere) (i) There is $N\in\mathcal{B}(\mathbb{R}^{n})$ with $\lambda_{n}(N)=0$ such that $f$ is twice differentiable at every $y\in\mathbb{R}^{n}\setminus N$.

2. (The Hessian is positive semidefinite) (ii) At every $y\in\mathbb{R}^{n}$ at which $f$ is twice differentiable one has $0_{n}\preceq D^{2}f(y)$.