The computation of (i) ε-kernels, (ii) approximate diameter, and (iii) approximate bichromatic closest pair are fundamental problems in geometric approximation. In each case the input is a set of points in d dimensions for constant d and an approximation parameter ε > 0. In this paper, we describe new algorithms for these problems, achieving significant improvements to the exponent of the ε-dependency in their running times, from roughly d to d / 2 for the first two problems and from roughly d / 3 to d / 4 for problem (iii). These results are all based on an efficient decomposition of a convex body using a hierarchy of Macbeath regions, and contrast to previous solutions that decomposed the space using quadtrees and grids. By further application of these techniques, we also show that it is possible to obtain near-optimal preprocessing time for the most efficient data structures for (iv) approximate nearest neighbor searching, (v) directional width queries, and (vi) polytope membership queries.