Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets

The unit distance graph $G_{\mathbb{R}^d}^1$ is the infinite graph whose nodes are points in $\mathbb{R}^d$, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version $G_{\mathbb{R}^2}^1$ of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of $G_{\mathbb{R}^d}^1$ to closed convex subsets $X$ of $\mathbb{R}^d$. We show that the graph $G_{\mathbb{R}^d}^1[X]$ is connected precisely when the radius of $r(X)$ of $X$ is equal to 0, or when $r(X)\geq 1$ and the affine dimension of $X$ is at least 2. For hyperrectangles, we give bounds for the graph diameter in the critical case that the radius is exactly 1.