Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity

We work in the setting of Euclidean Space and Real Matrices: Standing Notation, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers, the absolute value $|\cdot|$, the initial segments $[n]$, and Euclidean space $\mathbb{R}^{n}$ with its sum and difference of points, scalar multiples, dot product, Euclidean norm $\lVert\,\cdot\,\rVert$, distance $d_{E}$ and notion of openness, the real matrices, their differences and scalar multiples, the matrix-vector product, the identity matrix $I_{n}$ and the zero matrix $0_{n}$, the set $\mathcal{S}(n)$ of symmetric real $n\times n$ matrices, which contains $0_{n}$, the positive semidefinite ordering $\preceq$ and the norm $\lVert P\rVert$ of a symmetric real matrix, are all as fixed there. Since $0<2$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, where $2$ denotes $1+1$, the real number $\tfrac{s}{2}$, the product of $s$ with the multiplicative inverse of $2$, is defined for every $s\in\mathbb{R}$.

For an open set $V\subseteq\mathbb{R}^{n}$, that a function $\psi:V\to\mathbb{R}$ is of class $C^{2}$ on $V$ is understood in the sense of clause 3 of the definition of $C^{k}$ maps on a Euclidean open set; for such a $\psi$ and $z\in V$, $D\psi(z)\in\mathbb{R}^{n}$ is the gradient and $D^{2}\psi(z)$ the Hessian matrix, which lies in $\mathcal{S}(n)$ by claim 2 of Equality of Mixed Partial Derivatives, and Symmetry and Continuity of the Hessian.

Let $M\in\mathcal{S}(n)$, let $q\in\mathbb{R}^{n}$, let $c\in\mathbb{R}$, and let $Q:\mathbb{R}^{n}\to\mathbb{R}$ be given by

$$Q(z)=\tfrac{1}{2}\,z\cdot(Mz)+q\cdot z+c .$$

Then the following hold.

1. (Quadratic functions are of class $C^{2}$) (i) $Q$ is of class $C^{2}$ on $\mathbb{R}^{n}$, and $DQ(z)=Mz+q$ and $D^{2}Q(z)=M$ for every $z\in\mathbb{R}^{n}$. Moreover, for every open $V\subseteq\mathbb{R}^{n}$ the restriction of $Q$ to $V$ is of class $C^{2}$ on $V$, with gradient $Mz+q$ and Hessian $M$ at every $z\in V$.

2. (Translation) (ii) Let $V\subseteq\mathbb{R}^{n}$ be open, let $b\in\mathbb{R}^{n}$, and put $V-b=\{z\in\mathbb{R}^{n}:z+b\in V\}$. Then $V-b$ is open. If moreover $\psi:V\to\mathbb{R}$ is of class $C^{2}$ on $V$, then the function $\psi_{b}:V-b\to\mathbb{R}$ given by $\psi_{b}(z)=\psi(z+b)$ is of class $C^{2}$ on $V-b$, and $D\psi_{b}(z)=D\psi(z+b)$ and $D^{2}\psi_{b}(z)=D^{2}\psi(z+b)$ for every $z\in V-b$.

3. (Positive semidefinite quadratic forms are convex) (iii) Suppose $0_{n}\preceq M$ and let $C\subseteq\mathbb{R}^{n}$ be convex. Then the function $C\to\mathbb{R}$ whose value at $z$ is $Q(z)$ is convex on $C$.

4. (Quadratic perturbation of a semiconvex function) (iv) Let $C\subseteq\mathbb{R}^{n}$ be convex, let $\lambda\in\mathbb{R}$ satisfy $0\le\lambda$, and let $f:C\to\mathbb{R}$ be semiconvex on $C$ with constant $\lambda$. Then the function $g:C\to\mathbb{R}$ given by

$$g(z)=f(z)-\tfrac{1}{2}\,z\cdot(Mz)+q\cdot z+c$$

is semiconvex on $C$ with constant $\lambda+\lVert M\rVert$.