As part of a larger problem, I am working with a closed planar curve, and I am trying to show it is a circle. Consider a curve with the following properties, parameterized in polar coordinates $r(\theta), 0\leq\theta<2\pi$:
- The curve is strictly convex and $C^2$.
- I have derived a relationship between the curvature $\kappa(\theta)$ and $r(\theta)$ that says $\kappa$ decreases as $r$ increases, and vice versa. In particular, any local minimum of $r$ is a local maximum of $\kappa$.
I suspect/hope that the second bullet point forces $r$ to be constant. This suspicion comes from looking at ellipses. In this case, the two points on the ellipse that intersect the major axis are maximum points for $r$, and also maximum points for $\kappa$. Similarly, the minimum points of $r$ are also minimum points of $\kappa$.
Of course, in the general case it may not always be true that the minimum points of $r$ and $\kappa$ coincide, but I don't need that much. I only need to show in my situation, where every maximum of $r$ is a minimum of $\kappa$, that $r$ must be constant.