On the hyperbolicity of Delaunay triangulations

If X is a geodesic metric space and x1, x2, x3 is an element of X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is hyperbolic if there exists a constant delta >= 0 such that any side of any geodesic triangle in X is contained in the delta-neighborhood of the union of the two other sides. In this paper, we study the hyperbolicity of an important kind of Euclidean graphs called Delaunay triangulations. Furthermore, we characterize the Delaunay triangulations contained in the Euclidean plane that are hyperbolic.