The number of edges of many faces in a line segment arrangement

We show that the maximum number of edges bounding m faces in an arrangement of n line segments in the plane is O(m2/3n2/3+nα(n)+n logm). This improves a previous upper bound of Edelsbrunner et al. [EGS] and almost matches the best known lower bound which is Ω(m2/3n2/3 + nα(n)). In addition, we show that the number of edges bounding any m faces in an arrangement of n line segments with a total of t intersecting pairs is O(m2/3t1/3 +nα( t n)+nmin{logm, log t n}), almost matching the lower bound of Ω(m2/3t1/3 + nα( t n)) demonstrated in this paper.