Consider the following Theorem:
Theorem. Let $c\in\mathcal C^2([0,L],\mathbb R^2)$ be a simply (i.e. injective) closed (i. e. $c(0)=c(L)$) curve parametrized by arc length (i.e. $\|\dot c(t)\|=1$ for all $t$). Then the following two statements are equivalent:
- The curvature $\kappa(t):=\dfrac{\det(\dot c(t), \ddot c(t))}{\|\dot c(t)\|^3}$ is always non-negative or always non-positive.
- There is a convex set $S\subset\mathbb R^2$ such that $\partial S=c([0,L])$.
I found a proof of a variation of this Theorem where the second condition is replaced by supporing lines. However, I don't know how to prove that the Image of $c$ is the boundary of a convex set if and only if $c$ has a supporting line through each of its points.
Is there a "direct" way to prove the Theorem?
My ideas:
If $\kappa$ changes sign then we can have a look at where the sign changes and use the implicit function Theorem to construct a segment between two points on the curve that lies "outside" of the curve. However I have trouble formalising this argument.
If $\kappa$ doesn't change sign I don't know how to prove that the curve is the boundary of a convex set...