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The space of square-integrable functions $f:[0,1]\rightarrow\mathbb{R}$ is well conceivable: it's essentially an $\infty$-dimensional Euclidean space (the Hilbert space $L^2$) with well interpretable dimensions (= frequencies, modes).

But is there something like the space of Jordan curves $f:[0,1]\rightarrow\mathbb{R}^2$? One might guess it's $L^2 \times L^2$, but that's not true, since being a Jordan curve imposes stronger restrictions on $f$, so the sought space $\mathcal{J}$ is only a subset of $L^2 \times L^2$. But what kind of subset? How is it shaped? Is it a subspace? Of which dimension?

[For me this a general problem: to imagine and understand "spaces of shapes": what is a sensible metric (→ metric space), how - eventually - to add and scale shapes (→ vector space), what's the dimension, and how to interpret the dimensions?]

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Didn't you want to mean $\mathcal{J}$ as a subset of $C\times C$ where $C$ denotes the class of continuous functions $[0,1]\to\Bbb R$? as Jordan curves are continuous.. – Berci Feb 15 '13 at 1:39
Well, in particular, every continuous function from a compact space to a finite-dimensional vector space is $L^2$ ... – Neal Feb 15 '13 at 1:47
up vote 3 down vote accepted

If your motivation is geometric, it is natural to consider equivalence classes up to reparametrization. That is, to take the quotient of the set of homeomorphisms $f:S^1\to\mathbb R^2$ by the group of homeomorphisms $\psi:S^1\to S^1$, with the group action being composition ($f\sim f\circ \psi$). Of course, this moves the matter even further away from linear spaces. On the other hand, sensible metrics emerge, such as the simple Hausdorff metric and more sophisticated metrics which use the quotient structure. Efficient construction of geodesics between two given planar curves is a popular problem in this area.

The key word is shape space and while I do not know of a canonical reference for this subject, a simple search brings up a number of interesting articles.

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