The present paper sheds new light on ongoing drawbacks of three classical algorithms dedicated to initial contact detection between overlapping convex polyhedra, (i) Cundall's common plane, (ii) Nezami's fast common plane and (iii) Gilbert-Johnson-Keerthi's algorithm (GJK). Solutions to these drawbacks are suggested and implemented into revised versions of those three algorithms, which are further benchmarked for accuracy and speed using nine overlapping contact situations. The benchmarking results show that the revised version of GJK, called GJK T D, and Nezami (revised) return values of the contact normal components and overlap depth which are identical to machine precision, whereas Cundall (revised) results differ beyond the ninth decimal place. Furthermore, for a given contact situation, GJK-TD returns those values within a few tens of microseconds on average, whereas Nezami (revised) and Cundall (revised) are respectively 6 and 65 times more computationally intense. It is believed that the robustness and efficiency of GJK-TD will boost its use into DEM simulations, all the more that this versatile algorithm may easily be customized to detect contact between convex polyhedra and spheroid particles.