A Sum in Separated Variables of Semiconvex Functions is Semiconvex

Let $m\ge1$ and $n\ge1$ be natural numbers and let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure. For a natural number $p$ regard Euclidean space $\mathbb{R}^{p}$ as a real vector space, with the sum of points, the scalar multiple and the difference of points, and write $\lVert\,\cdot\,\rVert$ for the Euclidean norm, abbreviating $\lVert z\rVert^{2}=\lVert z\rVert\cdot\lVert z\rVert$. Let

$$\iota:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{m+n}$$

be the concatenation map, a bijection by claim 1 of that lemma.

Let $C_{1}\subseteq\mathbb{R}^{m}$ and $C_{2}\subseteq\mathbb{R}^{n}$ be convex, let $\mu_{1},\mu_{2}\in\mathbb{R}$ satisfy $0\le\mu_{1}$ and $0\le\mu_{2}$, and let $u_{1}:C_{1}\to\mathbb{R}$ be semiconvex on $C_{1}$ with constant $\mu_{1}$ and $u_{2}:C_{2}\to\mathbb{R}$ be semiconvex on $C_{2}$ with constant $\mu_{2}$. Let $\mu\in\mathbb{R}$ satisfy $\mu_{1}\le\mu$ and $\mu_{2}\le\mu$.

Put

$$C=\{\,\iota(\xi,\eta)\ :\ \xi\in C_{1},\ \eta\in C_{2}\,\}\subseteq\mathbb{R}^{m+n},$$

and let $w:C\to\mathbb{R}$ be the function determined by

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

which is well defined because $\iota$ is injective, so that each point of $C$ arises from exactly one such pair.

Then the following hold.

1. (The product is convex) $C$ is a convex subset of $\mathbb{R}^{m+n}$.

2. (Semiconvexity of the separated sum) $0\le\mu$, and $w$ is semiconvex on $C$ with constant $\mu$.