Improved lower bound towards Chen-Chv\'atal conjecture

We prove that in every metric space where no line contains all the points, there are at least $\Omega(n^{2/3})$ lines. This improves the previous $\Omega(\sqrt{n})$ lower bound on the number of lines in general metric space, and also improves the previous $\Omega(n^{4/7})$ lower bound on the number of lines in metric spaces generated by connected graphs.