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We know that in a lot of cases, the limit of a sequence of closed curves does not preserve perimeter. For instance, here is a false proof that $\pi=4$ which wrongly assumes this property (see this thread for what went wrong).

In the above example, we could define the sequence of polygons $(C_n)$ which successively approximate the half-unit circle, where $C_0$ is the unit square and $C_{n+1}$ is obtained by “removing the corners” of $C_n$. Now there is a sense in which the sequence does converge to the half-unit circle, call it $C$.

Since there is no natural extension of functional convergence to arbitrary curves, we can instead think about it in terms of infima on the distances between the curves. That is, we can see in this case that for every $\epsilon > 0$, there exists an $N$ such that whenever $n \geqslant N$, if $x \in C_n$ then $\inf\left\{y \in C\ |\ d(x,y)\right\} < \epsilon$.

Under this definition, we see clearly that while the sequence of polygons converges to a curve, the sequence of perimeters does not converge to the perimeter of the limit curve, hence the contradiction that $\pi=4$.

However in other cases, the so-defined limit does preserve perimeter. For example, consider the sequence $(P_n)$ of regular $n$-gons with apothem $a=1$. Then under the above definition, the sequence converges to the unit circle of perimeter $2\pi$. And the perimeter $\ell_n$ of each polygon $P_n$, given by $$\ell_n=2n \tan{\frac{\pi}{n}}$$ does conveniently converge to $2\pi$.

So my question is the following:

Let $(L_n)$ be a sequence of closed curves in $\mathbb{R}^2$ which converge to $L$ in the sense that $$\inf\left\{x \in L_n, y \in L\ |\ d(x,y)\right\} \to 0$$ as $n \to \infty$. Under what conditions should the sequence formed by the perimeters of $L_1$, $L_2,\ldots$ converge to the perimeter of $L$?

I would also like to ask for some reading material concerning perimeters and the limits of curves in Euclidian space, as I could not find a formal definition or treatment of such on the web. Further, is there an alternative definition of the limit of a curve to mine, for example one which does preserve perimeter?

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  • A nice sufficient condition for convergence is that the curves are all convex. Concerning the literature, try Falconer's book The geometry of fractal sets: despite the title, it deals with sets of integer dimension too, in particular curves. –  Mar 27 '16 at 06:29

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As stared, your definition of convergence "$\inf\left\{x \in L_n, y \in L\ |\ d(x,y)\right\} \to 0$" doesn't imply anything: any two curves with a common point satisfy it. You need to throw a supremum in there, as in the definition of Hausdorff distance.

A nice sufficient condition for convergence of length under the convergence in Hausdorff distance is that is that the curves are all convex (i.e., bound a convex region). For a proof, see the accessible introduction From Euclid to Alexandrov: a guided tour by Petrunin and Yashinski, namely Claim 2.11. (These are lecture notes for MASS program at PSU, i.e., aimed at strong undergraduates).

You can find more advanced literature by searching for some combinations of "rectifiable curves", "convergence" and "geometric measure theory". For example: