Bounds to the normal for proximity region graphs

In a proximity region graph G in Rd, two distinct points x,y of a point process μ are connected when the 'forbidden region' S(x,y) these points determine has empty intersection with μ. The Gabriel graph, where S(x,y) is the open disk with diameter the line segment connecting x and y, is one canonical example. When μ is a Poisson or binomial process, under broad conditions on the regions S(x,y), bounds on the Kolmogorov and Wasserstein distances to the normal are produced for functionals of G, including the total number of edges and the total length. Variance lower bounds, not requiring strong stabilization, are also proven to hold for a class of such functionals.