Curve with with curvature $k(s)\ge 1$ everywhere has diameter $\le 2$

Question

Let $\alpha(s)$ be a simple closed plane curve. Define the diameter $d_\alpha$ of $\alpha(s)$ to be $$d_\alpha = \sup_{t,s\in\mathbb R} \| \alpha(s) - \alpha(t) \|.$$ Assume the curvature $k(s)\ge 1$ for all $s$. Prove that $d_\alpha \le 2$ (or some other constant independent of $\alpha$; $2$ is the best possible constant).

While I know a few things about Riemannian manifolds, I don't know anything about the classical geometry of curves, so this question is bugging me. Because the total curvature around a simple smooth curve is $2\pi$, we know $\alpha(s)$ has length at most $2\pi$, so we can fit it in a circle of radius $2\pi$ and get a bound on the diameter that way. And while that bound can be refined a little, that crude method isn't good enough to get the bound down to $2$. How do we obtain the best possible constant? (Note that considering a circle of radius $1$ shows that $2$ is indeed the best possible bound.)

This is problem 3101 from Problems and Solutions in Mathematics (Major American Universities Ph.D. Qualifying Questions and Solutions - Mathematics) by Ta-Tsien Li (Editor).

The solution in the book claims that $\alpha(s)$ must be an "oval" and proceeds by explicit calculation from there. I don't know what an "oval" is, or why we can deduce that property of $\alpha(s)$ from the hypotheses.

Answer 1

The solution actually says:

From the hypothesis that $k(s) \geq 1$ for all $s$, we know that the simple closed curve $\alpha$ is an oval.

I think the mistake is thinking that an oval is a synonym for ellipse. It is actually more general. In the book Modern Differential Geometry of Curves and Surfaces with Mathematica you can see in Definition $6.25$ that:

An oval is a simple closed curve for which the signed curvature $\kappa2$ is always strictly positive or always strictly negative.

The problem requires that $k(s) > 1$, so $\alpha$ is an oval. The chapter then goes on to define the "pedal parametrization" in Definition $6.30$ which is the property that is used in the solution.