How do we known that an arc-length parametrization of a piecewise $C^1$-curve exists? I know that if a curve is regular (i.e. its velocity vectors are non-zero), then such a parametrization exists. But otherwise?
I asked this question since this is used in the proof of the isoperimetric inequality:
If $C$ is a simple closed piecewise $C^1$ curve of length $\ell$, with its interior having area $A$, then $\ell^2-4\pi A \geq 0$. Furthermore, equality holds if and only if $C$ is a circle.
A proof of isoperimetric inequality as presented in do Carmo's book "Differenatial Geometry of Curves and Sufaces" requires choosing an arc-length parametrization for the simple closed curve under consideration. (See also here: A proof of the Isoperimetric Inequality - how does it work? for the proof I have in mind, although the arc-length parametrization is needed for the part which says that an equality holds if and only if the curve is a circle, which is not discussed in the linked post.)