The specific case of a Delaunay triangulation (DT) satisfies many additional desirable criteria , several of which also serve as independent definitions of the DT . Of particular relevance here, for a grid of points G and function evaluations F , the DT minimizes the area (volume for d > 2) of the polyhedral surface representing F -an illustration of this condition is shown in Fig. (left). The DT also minimizes the size of the largest open ball (b(x; d) = {y ∈ R N : |y -x| < d}) which bounds a simplex and thus avoids large simplices corresponding to large neighbourhoods -this is also shown in Fig. (lower right).