On 3-decomposable geometric drawings of $K_n$

The point sets of all known optimal rectilinear drawings of $K_n$ share an unmistakeable clustering property, the so--called {\em 3--decomposability}. It is widely believed that the underlying point sets of all optimal rectilinear drawings of $K_n$ are 3--decomposable. We give a lower bound for the minimum number of $(\le k)$--sets in a 3--decomposable $n$--point set. As an immediate corollary, we obtain a lower bound for the crossing number $\rcr(\dd)$ of any rectilinear drawing $\dd$ of $K_n$ with underlying 3--decomposable point set, namely $\rcr(\dd) > {2/27}(15-\pi^{2})\binom{n}{4}+\Theta(n^{3}) \approx 0.380029\binom{n}{4} + \Theta(n^3)$. This closes this gap between the best known lower and upper bounds for the rectilinear crossing number $\rcr(K_n)$ of $K_n$ by over 40%, under the assumption of 3--decomposability.