Before revision:  (C) We show a construction of weak } -nets which have the following additional property: For a heavy body, there is a net point that stabs the body, and it is also a good centerpoint for the points contained inside the body. (D) We present a variant of a known algorithm for approximating a centerpoint, improving the running time from (d^9) to (d^7).
After revision:  (C) We show a construction of weak }  (d^9) to (d^7).