I have been reading about the 4 vertices theorem in differential geometry, I have a doubt, we know that the curvature of a convex curve fulfills the following:
$\kappa \ \text{is periodic}$
$\kappa \ \text{has at least 4 critical points}$
$\kappa\geq0$
$\int \kappa=2\pi$
The question I have is, are these curvature conditions sufficient for the curve to necessarily be convex? or is there a curve that serves to show that no, I would appreciate it if you could tell me. First of all, Thanks