Concatenation and the Sup-Convolution of a Sum in Separated Variables

Throughout we work in the setting of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation, whose notation is in force in the dimensions $m$, $n$ and $N=m+n$, for natural numbers $m$ and $n$ with $1\le m$ and $1\le n$. In particular $\iota:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{N}$ is the concatenation map, a bijection; we abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\cdot\lVert z\rVert$. By claims 2, 3 and 4 of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space the map $\iota$ is linear and satisfies

for all $\xi,\xi'\in\mathbb{R}^{m}$ and $\eta,\eta'\in\mathbb{R}^{n}$.

Let $u_{1}:\mathbb{R}^{m}\to\mathbb{R}$ and $u_{2}:\mathbb{R}^{n}\to\mathbb{R}$ be functions whose sets of values have upper bounds $C_{1}$ and $C_{2}$ in $\mathbb{R}$, and let $w:\mathbb{R}^{N}\to\mathbb{R}$ be the function determined by

which is well defined and defined at every point of $\mathbb{R}^{N}$ because $\iota$ is a bijection. Let $\lambda\in\mathbb{R}$ satisfy $0<\lambda$. Then the following hold.

1. (Origins and coordinatewise convergence) ¶ $\iota\bigl(0_{\mathbb{R}^{m}},0_{\mathbb{R}^{n}}\bigr)=0_{\mathbb{R}^{N}}$. Moreover, let $(\xi_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{m}$, let $(\eta_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{n}$, let $\xi\in\mathbb{R}^{m}$ and let $\eta\in\mathbb{R}^{n}$. Then $\bigl(\iota(\xi_{k},\eta_{k})\bigr)_{k\in\mathbb{N}}$ converges to $\iota(\xi,\eta)$ in $\mathbb{R}^{N}$ if and only if $(\xi_{k})_{k\in\mathbb{N}}$ converges to $\xi$ in $\mathbb{R}^{m}$ and $(\eta_{k})_{k\in\mathbb{N}}$ converges to $\eta$ in $\mathbb{R}^{n}$.

2. (The sum is bounded above) ¶ The real number $C_{1}+C_{2}$ is an upper bound for the set of values of $w$. Consequently the sup-convolutions $u_{1}^{\lambda}$, $u_{2}^{\lambda}$ and $w^{\lambda}$ of $u_{1}$, $u_{2}$ and $w$ with parameter $\lambda$ are all defined.

3. (The sup-convolution separates) ¶ For all $\xi\in\mathbb{R}^{m}$ and $\eta\in\mathbb{R}^{n}$,