Looking at a separate question here, I came upon a statement which I suspect to be true. However, an initial search did not reveal anything on the subject.
Let $P$ be a differentiable path of length $L$ in $\mathbb R^2$. That is, $P$ is the image of $f:[0, L]\to\mathbb R^2$, where $f$ is a differentiable mapping and $|f'|=1$ everywhere.
My claim is that there exists a closed ball $B=\{x\in\mathbb R^2: |x-x_0|\le r\}$ such that $P\subseteq B$ and $r\le\frac L2$.
I haven't been able to construct a counterexample, but a statement as simple as this has undoubtedly already been addressed. The weaker restriction $r\le\frac L{\sqrt{3}}$ can be derived from Jung's Inequality, but I have not found anything addressing path length specifically.
Has this theorem been proven, or does a counterexample exist? If it is true, does it extend to $\mathbb R^n$? Any input would be appreciated.
Such a ball is very easy to find - we can in fact always take $x_0 = f(L/2), r= L/2$. This works because the curve is unit-speed: for any $t \in [0,L]$ we have $$|f(t) - f(L/2)| \le \sup_t(|f'(t)|)\cdot\left|t - \frac L2\right|,$$ so since $|f'| = 1$ and $t\in[0,L]$ we conclude $|f(t) - f(\frac L2)|\le \frac L2.$