The purpose of this paper is to generalize the convexity of Mabuchi's
functional to the conic setting. We first established a frame to study conic
cscK metrics, and then the conic Mabuchi functional was introduced in such a
way that conic cscK metrics are its critical points. Finally we proved the
convexity of the conic Mabuchi functional along any conic geodesics.Comment: Some errors and typos are corrected, and one Appendix is added to
  explain a new regularization technique for weak conic geodesic