Before revision:  We also prove that any such net must have size at least \Omega(1/}\eps As a concrete example, in three dimensions all \eps-heavy bodies in [0,1]^3 can be stabbed by   (1/\eps^{2/3}) lines.
After revision:  We also prove that any such net must have size at least \Omega(1/}\eps As a concrete example, in three dimensions all \eps-heavy bodies in [0,1]^3 can be stabbed by  \Theta (1/\eps^{2/3}) lines.