Most well known densities satisfy thier condition. This seems to require much more computation, in the form of multiplications, than simply finding the extreme points in the directions of step 1 above. Let N be the number of points left after step 2. Divide and conquer for linear expected time . 0 denotes an arbitrarily small but fixed constant.
Further improvements seem to require new ideas.
J. L. Bentley and M. I. Shamos. It is an open problem whether we can do better than this in three and higher dimensions. J.L. Bentley and M. Shamos. Divide and conquer for linear expected time . J.L. Bentley and M.I. Shamos.