New lower bounds for the number of (≤ k)-edges and the rectilinear crossing number of K_n

We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ ⌊(n-2)/2⌋ the number of (≤ k)-edges is at least equation presented, improves the previous best lower bound in [12]. 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 (41/108+ε(n/4) + O(n³) ≥ 0.379688 (n/4) + O(n³), which improves the previous bound of 0.37533 (n/4) + O(n³) in [12] and approaches the best known upper bound 0.380559 (n/4) + Θ(n³) in [4]. The proof is based on a result about the structure of sets attaining the rectilinear crossing number, for which we show that the convex hull is always a triangle. Further implications include improved results for small values of n. We extend the range of known values for the rectilinear crossing number, namely by cr̄(K₁₉)=1318 and cr̄(K₂₁)=2055. Moreover, we provide improved upper bounds on the maximum number of halving edges a point set can have.