Before revision:  (D) We present a variant of a known algorithm for approximating a centerpoint, improving the running time from (d^9) to (d^7). Our analysis of this algorithmis arguably cleaner than the previous version .
After revision:  (D) We present a variant of a known algorithm for approximating a centerpoint, improving the running time from (d^9) to progress on this well known problem in over twenty years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm .