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I posted this question a few days ago to https://or.stackexchange.com/questions/13173/optimization-over-loop-spaces but didn't receive any replies, so I thought I would try here (if this is improper then I apologize!):

Has anyone written any deep work about infinite-dimensional optimization over "loop spaces", i.e. the space of simple closed curves in (say) $\mathbb{R}^2$? Of course, one could simply consider the parameterization $f(t):[0,1]\to\mathbb{R}^2$ subject to the constraint that $f(0)=f(1)$, but I'm curious if there is anything we can say about regarding these objects independently of their parameterization; for instance, the Euclidean TSP on points $\{p_1,\dots,p_n\}$ becomes $\min_L\mathrm{length}(L)$ subject to the constraint that $p_i\in L$ for all $i$. A related concept would be taking the set of convex shapes in the plane, with addition defined via Minkowski sums, but things clearly become more challenging when one relaxes the convexity requirement.

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    $\begingroup$ Sure, there is a significant literature on this problem. Typically the literature is in relatively concrete contexts, like finding critical points of the length functional, or elastic bending energy, etc. Asked in this generality you're basically talking about finding critical points of arbitrary functions on arbitrary manifold domains, so there's some fairly hard constraints on what kind of theorems you can prove. $\endgroup$ Commented May 22 at 17:08
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    $\begingroup$ Significant parts of geometric measure theory can be traced back to the question "can we minimize a functional on manifolds in a parametrisation invariant sense". But when restricting to 1d-loops, there is no real downside to using some sort of arc-length or similar parametrisation and taking the Calc-var route. And in any case, the main thing one obtains are conditions for local minima, which are just the equations for some sort of geodesic. One can then vary the end-points, but that is just a standard finite dimensional problem. $\endgroup$ Commented May 22 at 17:21

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Not sure if you have something like this in mind, but there is the work on Repulsive Curves by Chris Yu, Henrik Schumacher, and Keenan Crane. There is also a short conference talk about this and a longer YouTube video here.

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  • $\begingroup$ That's a great example! Thank you. $\endgroup$ Commented May 23 at 18:10

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