Reduction of the Theorem on Sums to a Global Quadratic Bound

Throughout we work in the setting of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation on which it rests, is in force in the dimensions $n_{1}$, $n_{2}$ and $N=n_{1}+n_{2}$, for natural numbers $n_{1}$ and $n_{2}$ with $1\le n_{1}$ and $1\le n_{2}$. In particular $\iota:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\to\mathbb{R}^{N}$ is the concatenation map, a bijection. Closed balls $\bar{B}(x,\rho)$ of $(\mathbb{R}^{q},d_{E})$ are those of Closed Ball in a Metric Space.

We abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\cdot\lVert z\rVert$, and $\tfrac{s}{2}$ denotes the product of $s\in\mathbb{R}$ with the multiplicative inverse of $2=1+1$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field. The sets $\mathbb{R}^{n_{1}}$, $\mathbb{R}^{n_{2}}$ and $\mathbb{R}^{N}$ are open, directly from Open Subset of a Metric Space. That a quadruple is approximable by test data from above for a function on an open set is as defined there.

Let $\Omega_{1}\subseteq\mathbb{R}^{n_{1}}$ and $\Omega_{2}\subseteq\mathbb{R}^{n_{2}}$ be open and, for $i\in\{1,2\}$, let $u_{i}:\Omega_{i}\to\mathbb{R}$ be upper semicontinuous on $\Omega_{i}$. Put

$$\Omega=\{\,\iota(\xi,\eta)\ :\ \xi\in\Omega_{1},\ \eta\in\Omega_{2}\,\},$$

which is open in $\mathbb{R}^{N}$ by Twice Differentiability of a Sum in Separated Variables (i), and let $w:\Omega\to\mathbb{R}$ be the function determined by

$$w\bigl(\iota(\xi,\eta)\bigr)=u_{1}(\xi)+u_{2}(\eta)\qquad(\xi\in\Omega_{1},\ \eta\in\Omega_{2}),$$

which is well defined because $\iota$ is injective.

Let $V\subseteq\mathbb{R}^{N}$ be open with $\Omega\subseteq V$, let $\varphi:V\to\mathbb{R}$ be of class $C^{2}$ on $V$, and let $\hat{x}\in\Omega$ be a point at which $w-\varphi$ has a local maximum relative to $\Omega$. Let $(\hat{x}_{1},\hat{x}_{2})\in\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$ be the unique pair with $\iota(\hat{x}_{1},\hat{x}_{2})=\hat{x}$, and let $(p_{1},p_{2})$ be the unique pair with $\iota(p_{1},p_{2})=D\varphi(\hat{x})$; put $A=D^{2}\varphi(\hat{x})\in\mathcal{S}(N)$. Note that $\hat{x}_{i}\in\Omega_{i}$ for $i\in\{1,2\}$, since $\hat{x}\in\Omega$ and $\iota$ is injective.

Let $\theta\in\mathbb{R}$ be positive and put $A_{\theta}=A+\theta I_{N}$, which lies in $\mathcal{S}(N)$ by Euclidean Space and Real Matrices: Standing Notation (iii).

For $i\in\{1,2\}$ put $\Omega_{i}-\hat{x}_{i}=\{\zeta\in\mathbb{R}^{n_{i}}:\zeta+\hat{x}_{i}\in\Omega_{i}\}$, which is open by Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity (ii) and contains $0_{\mathbb{R}^{n_{i}}}$, and let $\tilde{u}_{i}:\Omega_{i}-\hat{x}_{i}\to\mathbb{R}$ be given by

$$\tilde{u}_{i}(\zeta)=u_{i}(\zeta+\hat{x}_{i})-p_{i}\cdot\zeta-u_{i}(\hat{x}_{i}),$$

so that $\tilde{u}_{i}(0_{\mathbb{R}^{n_{i}}})=0$.

Then there are a positive $r\in\mathbb{R}$ and functions $v_{1}:\mathbb{R}^{n_{1}}\to\mathbb{R}$ and $v_{2}:\mathbb{R}^{n_{2}}\to\mathbb{R}$ for which the following hold.

1. (Localised summands) (i) For each $i\in\{1,2\}$: the closed ball $\bar{B}(0_{\mathbb{R}^{n_{i}}},r)$ is contained in $\Omega_{i}-\hat{x}_{i}$; the function $v_{i}$ is upper semicontinuous on $\mathbb{R}^{n_{i}}$; the set of values of $v_{i}$ has an upper bound in $\mathbb{R}$; $v_{i}(\zeta)=\tilde{u}_{i}(\zeta)$ for every $\zeta\in\mathbb{R}^{n_{i}}$ with $\lVert\zeta\rVert\le r$; and in particular $v_{i}(0_{\mathbb{R}^{n_{i}}})=0$.

2. (Global quadratic bound) (ii) For all $\xi\in\mathbb{R}^{n_{1}}$ and $\eta\in\mathbb{R}^{n_{2}}$,

$$v_{1}(\xi)+v_{2}(\eta)\ \le\ \tfrac{1}{2}\,\iota(\xi,\eta)\cdot\bigl(A_{\theta}\,\iota(\xi,\eta)\bigr).$$

3. (Test data returns to the original functions) (iii) Let $i\in\{1,2\}$ and let $X\in\mathcal{S}(n_{i})$. If the quadruple $\bigl(0_{\mathbb{R}^{n_{i}}},v_{i}(0_{\mathbb{R}^{n_{i}}}),0_{\mathbb{R}^{n_{i}}},X\bigr)$ is approximable by test data from above for $v_{i}$, the domain being $\mathbb{R}^{n_{i}}$, then the quadruple $\bigl(\hat{x}_{i},u_{i}(\hat{x}_{i}),p_{i},X\bigr)$ is approximable by test data from above for $u_{i}$, the domain being $\Omega_{i}$.