Four-Vertex Theorems, Sturm Theory and Lagrangian Singularities

We prove that the vertices of a curve γ⊂R ⁿ are critical points of the radius of the osculating hypersphere. Using Sturm theory, we give a new proof of the (2k+2)-vertex theorem for convex curves in the Euclidean space R ^2k . We obtain a very practical formula to calculate the vertices of a curve in R ⁿ . We apply our formula and Sturm theory to calculate the number of vertices of the generalized ellipses in R ^2k . Moreover, we explain the relations between vertices of curves in Euclidean n-space, singularities of caustics and Sturm theory (for the fundamental systems of solutions of disconjugate homogeneous linear differential operators L:C ^∞(S ¹)→C ^∞(S ¹)).