Improved lower bound towards Chen-Chvátal conjecture

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