Geometric drawings of with few crossings

We give a new upper bound for the rectilinear crossing number cr ¯ ( n ) of the complete geometric graph K n . We prove that cr ¯ ( n ) ⩽ 0.380559 ( n 4 ) + Θ ( n 3 ) by means of a new construction based on an iterative duplication strategy starting with a set having a certain structure of halving lines.