Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets

The unit distance graph G1Rd is the infinite graph whose nodes are points in R, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version G1R2 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 G1Rd to closed convex subsets X of R. We show that the graph G1Rd [X] is connected precisely when the radius of r(X) of X is equal to 0, or when r(X) ≥ 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.