In the book "E. M. Stein and R. Shakarchi, Fourier Analysis: an introduction. Princeton University Press, 2003", the author proved the isoperimetric inequality using Fourier series. However, I found out something absurd in the proof. The following is the statement in the book.

However, the author started the proof by stating that:enter image description here

According to my understanding, in general, we may not be able to impose an arc-length parametrization. For example, when a curve is regular, then we can impose an arc-length parametrization on the curve. Certainly, the proof of this requires the fact that the curve is regular. Therefore, I suspect that we may not be able to impose a arc-length parametrization and thus the assumption in the proof in the second picture may not be valid for general simple, closed curve, and thus we may need to add extra assumption on the curve $\Gamma$ (say, regularity). Any suggestion on this??


Yes and no.

I just had a look at the discussion in the book you cited. If you check you'll see at the beginning of the section that they do assume $\gamma$ to be of class $C^1$, in which case the statement is not a problem.

Admittedly this is a bit unsatisfactory, but you should keep in mind that the book is intended to be an introduction. If you look at the similar discussion in volume three of the series ('Real Analysis', same authors) you'll find an in-depth discussion about the relationship between rectifiability, differentiability and the existence of arclength parametriziation of (rectifiable) curves. Briefly, a rectifiable curve is a curve $z(t), a<t<b$ for which the following supremum is finite $$\sup \sum |z(t_j) - z(t_{j-1})|$$ where the $\sup$ is taken over all partitions $ a = t_0< t_1 <\ldots< t_n$. That supremum is then called the length $\ell$ of the curve.

One can show (and in the book I mentioned it is shown) that such curves have coordinate functions which are of bounded variation, which means they are differentiable almost everywhere. It is also shown that the length of the curve can be expressed by $$\ell = \int \left( (z_2^\prime)^2 + (z_2^\prime)^2 \right)^{\frac{1}{2} }dt $$ if and only if the parametrization is absolutely continuous (note though, that the integral is also well defined if the functions are only of bounded variation).

Finally they show that for rectifiable curves you can always define an arclength parametrization, so that this formula can be applied. Starting from this formula you can do the whole discussion rigorously, but, as I tried to indicate, this is not completely trivial. I guess this is why they restricted themselves to $C^1$ in the first discussion.

Apart from that my experience is that it is often difficult -- or at least hard work -- to follow proof in Steins' book in detail, but it's extremely hard to find errors which are that easy to spot.


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