Tripod Configurations

In this paper, we research the tripod configurations. By Serge Tabachnikov, see Theorem 2 of Appendix [1] says that for any smooth convex closed curve, there exist at least two tripod configurations. In this paper we want to use another way to construct tripod configurations. Use a intuitive way by a geometrical approach to construct it. We use minimax method, and do some deformation such that the distance of the Y-shaped will decrease, but not all of the Y-shaped will degenerate, it will converge to a critical point which will not degenerate, and we explain that this critical point is our tripod configuration.