The positive center is defined by Gage, which is to simplify the proof of the Gage inequality:

where is the curvature of the convex curve , and and are the length and the enclosed area of , respectively. Following Gage’s idea, Kaiser introduced the -positive center and put forward two conjectures by some computer graphics.  In this talk, we introduce some recent work about the positive center and the -positive center, and their applications in geometric inequalities. This is joint work with Pro. Shengliang Pan (Tongji University) and Dr. Pingliang Huang (Shanghai  University).