A Semiconvex Function of Class $C^2$ has Hessian Bounded Below by $-\lambda I_n$

Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers. Let $U\subseteq\mathbb{R}^n$ be an open and convex subset of Euclidean space $\mathbb{R}^n$, a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, let $\lambda\in\mathbb{R}$ satisfy $0\le\lambda$, and let $f:U\to\mathbb{R}$ be of class $C^2$ on $U$ and semiconvex on $U$ with constant $\lambda$.

Write $I_n$ for the identity matrix of size $n$ and $\mu M$ for the scalar multiple of a real matrix. The entries of $(-\lambda)I_n$ satisfy $\bigl((-\lambda)I_n\bigr)_{ij}=\bigl((-\lambda)I_n\bigr)_{ji}$, so that matrix is symmetric, and the Hessian matrix $D^2f(x)$ is symmetric by claim 2 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian; both therefore lie in the set $\mathcal{S}(n)$ of symmetric real $n\times n$ matrices, so the positive semidefinite ordering $\preceq$ applies to them.

Then for every $x\in U$,

$$(-\lambda)I_n\preceq D^2f(x).$$