Alexandrov's Theorem for Semiconvex Functions on an Open Convex Set

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 and finite index sets, the Euclidean norm $\lVert\,\cdot\,\rVert$, dot product, distance $d_{E}$, notion of openness and closed balls $\bar{B}(x,r)$, the Lebesgue measure $\lambda_{n}$, the notion of a null subset, the constant $\sigma_{n}$ with $\sigma_{n}^{2}=n$, and the conventions on Lipschitz maps are as fixed there. Write $\mathcal{S}(n)$ for the set of symmetric real $n\times n$ matrices, $I_{n}$ for the identity matrix, $\lambda A$ for the scalar multiple of a real matrix, and $\preceq$ for the positive semidefinite ordering on $\mathcal{S}(n)$; the matrix $(-\mu)I_{n}$ below belongs to $\mathcal{S}(n)$ because all of its off-diagonal entries vanish.

Let $U\subseteq\mathbb{R}^{n}$ be open and convex, let $\mu\in\mathbb{R}$ with $0\le\mu$, and let $f:U\to\mathbb{R}$ be semiconvex on $U$ with constant $\mu$. 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) The set of those $y\in U$ at which $f$ is not twice differentiable is null.

2. (Lower bound on the Hessian) (ii) At every $y\in U$ at which $f$ is twice differentiable one has

$$(-\mu)I_{n}\preceq D^{2}f(y).$$