Mollification Preserves Semiconvexity

Let $n$ be a natural number with $1\le n$, let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure, write $s<t$ to mean that $s\le t$ and $s\ne t$, let $|s|$ be the absolute value of $s$, and let $d_{\mathbb{R}}$ be the metric on $\mathbb{R}$ of The Absolute Value Metric on the Real Numbers. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, with the sum of points and the scalar multiple, and write $x-y$ for the difference of points. Let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on $\mathbb{R}^{n}$ and let $d$ be the Euclidean distance, a metric on $\mathbb{R}^{n}$; by The Open Subsets of a Metric Space Form a Topology the subsets open in $(\mathbb{R}^{n},d)$ form a topology. Write $\bar B(x,r)$ for the closed ball in $(\mathbb{R}^{n},d)$ and let $\lambda_{n}$ be Lebesgue measure on the Borel $\sigma$-algebra of $\mathbb{R}^{n}$.

Let $\Omega\subseteq\mathbb{R}^{n}$ be open in $(\mathbb{R}^{n},d)$ and convex, let $f:\Omega\to\mathbb{R}$ be continuous on $\Omega$ as a map from $(\mathbb{R}^{n},d)$ into $(\mathbb{R},d_{\mathbb{R}})$, let $\mu\in\mathbb{R}$ satisfy $0\le\mu$, and suppose that $f$ is semiconvex on $\Omega$ with constant $\mu$.

Let $\delta\in\mathbb{R}$ with $0<\delta$ and let $\rho$ be a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$; being smooth it is continuous on $\mathbb{R}^{n}$ by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and it vanishes at every $y$ with $\delta<\lVert y\rVert$, so the convolution $f*\rho$ is defined on

$$\Omega^{\delta}=\{x\in\mathbb{R}^{n}:\bar B(x,\delta)\subseteq\Omega\},$$

a set open by claim 2 of The $\delta$-Interior of an Open Subset of $\mathbb{R}^n$ is Open and convex by The $\delta$-Interior of a Convex Subset of $\mathbb{R}^n$ is Convex.

Then $f*\rho$ is semiconvex on $\Omega^{\delta}$ with constant $\mu$.