  We study metrics on conic 2-spheres when no Einstein metrics exist. In
particular, when the curvature of a conic metric is positive, we obtain the
best curvature pinching constant. We also show that when this best pinching
constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff
limit. This is a generalization of the previous results of Chen-Lin and
Bartolucci for 2-spheres with one or two conic points.
