Lavinia, just to make sure that both of us are talking about the same thing, let me state the theorem from Do Carmo:
The Four-Vertex Theorem: A simple closed convex curve has at least four vertices.
Actually, I can kind of see that for a convex curve this line L should divide the curve in two...
Lavinia, I have attached a scan of that page containing the proof in Do Carmo's book. I understand the lemma. I also understand the first para of the proof of the theorem. I am a bit lost after that. First, I don't understand this:
By convexity, and since p, q, r are distinct points on C, the...
I just began to read Do Carmo's Differential Geometry book and having breezed through most of the chapter 1, I am finding it difficult to see how the four vertex theorem for a convex figure is proved?
Could someone please provide a simple and easy to understand proof?