  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.
