On $(\le k)$-edges, crossings, and halving lines of geometric drawings of Kn

Let P be a set of points in general position in the plane. Join all pairs of points in P with straight line segments. The number of segment-crossings in such a drawing, denoted by cr(P ), is the rectilinear crossing number of P . A halving line of P is a line passing though two points of P that divides the rest of the points of P in (almost) half. The number of halving lines of P is denoted by h(P ). Similarly, a kedge, 0 ≤ k ≤ n/2− 1, is a line passing through two points of P and leaving exactly k points of P on one side. The number of (≤ k)-edges of P is denoted by E≤k(P ). Let cr(n), h(n), and E≤k(n) denote the minimum of cr(P ), the maximum of h(P ), and the minimum of E≤k(P ), respectively, over all sets P of n points in general position in the plane. We show that the previously best known lower bound on E≤k(n) is tight for k < ⌈(4n − 2)/9⌉ and improve it for all k ≥ ⌈(4n − 2)/9⌉. This in turn improves the lower bound on cr(n) from 0.37968 (n 4 ) +Θ(n3) to 277 729 (n 4 ) +Θ(n3) ≥ 0.37997 (n 4 ) +Θ(n3). We also give the exact values of cr(n) and h(n) for all n ≤ 27. Exact values were known only for n ≤ 18 and odd n ≤ 21 for the crossing number, and for n ≤ 14 and odd n ≤ 21 for halving lines. 2010 AMS Subject Classification: Primary 52C30, Secondary 52C10, 52C45, 05C62, 68R10, 60D05, and 52A22.