Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets

The unit distance graph G¹_Rd 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 G¹_R2 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 G¹_Rd to closed convex subsets X of R^d. We show that the graph G¹_Rd[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.