Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets

Abstract

The unit distance graph $G^1_{R^d}$ is the infinite graph whose nodes are points in $R^d$, with an edge between two points if the Euclidean distance between these points is $1$. The 2-dimensional version $G^1_{R^2}$ 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 $G^1_{R^d}$ to closed convex subsets $X$ of $R^d$. We show that the graph $G^1_{R^d}[X]$ 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.