The $\delta$-Interior of a Convex Subset of $\mathbb{R}^n$ is Convex

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, and write $s<t$ to mean that $s\le t$ and $s\ne t$. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, with the sum of points and the scalar multiple. Let $d$ be the Euclidean distance, a metric on $\mathbb{R}^{n}$, and write $\bar B(x,r)$ for the closed ball of centre $x$ and radius $r$ in $(\mathbb{R}^{n},d)$.

Let $\Omega\subseteq\mathbb{R}^{n}$ be convex, let $\delta\in\mathbb{R}$ with $0<\delta$, and put

Then $\Omega^{\delta}$ is convex.