  We provide a new lower bound on the number of $(\leq k)$-edges of a set of
$n$ points in the plane in general position. We show that for $0 \leq k
\leq\lfloor\frac{n-2}{2}\rfloor$ the number of $(\leq k)$-edges is at least $$
E_k(S) \geq 3\binom{k+2}{2} + \sum_{j=\lfloor\frac{n}{3}\rfloor}^k (3j-n+3), $$
which, for $k\geq \lfloor\tfrac{n}{3}\rfloor$, improves the previous best lower
bound in [J. Balogh, G. Salazar, Improved bounds for the number of ($\leq
k$)-sets, convex quadrilaterals, and the rectilinear crossing number of $K_n$].
As a main consequence, we obtain a new lower bound on the rectilinear crossing
number of the complete graph or, in other words, on the minimum number of
convex quadrilaterals determined by $n$ points in the plane in general
position. We show that the crossing number is at least $$
\Bigl({41/108}+\epsilon \Bigr) \binom{n}{4} + O(n^3) \geq 0.379631 \binom{n}{4}
+ O(n^3), $$ which improves the previous bound of $0.37533 \binom{n}{4} +
O(n^3)$ in [J. Balogh, G. Salazar, Improved bounds for the number of ($\leq
k$)-sets, convex quadrilaterals, and the rectilinear crossing number of $K_n$]
and approaches the best known upper bound $0.38058\binom{n}{4}$ in [O.
Aichholzer, H. Krasser, Abstract order type extension and new results on the
rectilinear crossing number].
