Hessian Lower Bound for a $C^2$ Test Function Touching a Semiconvex Function from Above

Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, with the sum of points, the scalar multiple, and the difference $x-y$ and dot product $x\cdot y$ of points; write $\lVert\,\cdot\,\rVert$ for the Euclidean norm, and let $d$ be the Euclidean distance, a metric on $\mathbb{R}^{n}$; a subset of $\mathbb{R}^{n}$ is open if and only if it is open in $(\mathbb{R}^{n},d)$, by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$.

Let $\Omega\subseteq\mathbb{R}^{n}$ be convex, let $\mu\in\mathbb{R}$ satisfy $0\le\mu$, and let $w:\Omega\to\mathbb{R}$ be semiconvex on $\Omega$ with constant $\mu$. Let $U\subseteq\Omega$ be open, let $x_{0}\in U$, let $\varphi:U\to\mathbb{R}$ be of class $C^{2}$ on $U$ (via clause 3 there), and suppose that the function $U\to\mathbb{R}$ whose value at $x$ is $w(x)-\varphi(x)$ has a local maximum at $x_{0}$ relative to $U$.

Write $D^{2}\varphi(x_{0})$ for the Hessian matrix of $\varphi$ at $x_{0}$, write $I_{n}$ for the identity matrix of size $n$, and write $\alpha M$ for the scalar multiple of a real matrix $M$ by $\alpha\in\mathbb{R}$. The entries of $(-\mu)I_{n}$ satisfy $\bigl((-\mu)I_{n}\bigr)_{ij}=\bigl((-\mu)I_{n}\bigr)_{ji}$, so that matrix is symmetric, and $D^{2}\varphi(x_{0})$ 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

$$(-\mu)I_{n}\preceq D^{2}\varphi(x_{0}).$$