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this question may be a bit sloppy on my part but I will make it anyway, I recently have been fascinated by the idea that the surface of a soap bubble film restricted to a boundary will be so as to minimize surface area, and I have been thinking if there is no such equivalent thing for a curve in $\mathbb{R}^2$, by that i mean something like given a finite set of points, that will work as if they were the boundary of the tridimensional case, to find a curve that will minimize the arc length of the path.

I do realize that given that the smallest distance between two points is a straight line and because the curve has to pass trough a set of points, comparing the length of the sections of the possible curves in between two points and continuing two by two points until the last one, each time the straight line will win in least length, and so a linear interpolation will be the curve with least arc length, but given that linear interpolation will in the majority of cases produce a path that is non-differentiable on a given set of points, I then thought of limiting this curves by only differentiable ones, but I suspect that only differentiability is not enough to guarantee the existence of such a curve, for i can take the linear interpolation curve, let's call it $f(x)$ and pass it trough a "differentiability filter" such as :

$$\frac{1}{2\varepsilon}\int^{x+\varepsilon}_{x-\varepsilon}f(t)dt = g_{\varepsilon}(x)$$

and the function $g_\varepsilon(x)$ will be equal to the function $f(x)$ outside of open balls of radii $\varepsilon$ from the points of non-differentiability and as $\varepsilon$ can be made as small as wanted it probably means there is no lower bound inside the set of differentiable functions.

So instead of only requiring differentiability and a minimum arc length if it was request that the function be of class $C^\infty$ and have minimum arc length, would there exist a solution ? and if so, is there some type of interpolation that accomplishes just that ? or at least is there some type of interpolation that shares some type of link with the soap bubble film in tree dimensions so aforementioned, something like least curvature or other properties? because it feels as if it should exist!

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  • $\begingroup$ You've made a very nice analysis of the differentiable case. Now notice that your filter is equivalent to convolving $f(x)$ with a box function $h(x)$ which is $1/(2\epsilon)$ when $|x|\le\epsilon$ and $0$ otherwise. If you replace $h$ with a Gaussian function of standard deviation $\epsilon$, then the filtered version of $f$ will be $C^\infty$. $\endgroup$ – Rahul Jan 28 '18 at 8:06
  • $\begingroup$ Splines under tension might be relevant. $\endgroup$ – marty cohen Jan 31 '18 at 14:21
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Perhaps what you are looking for is mean curvature flow (https://en.wikipedia.org/wiki/Mean_curvature_flow), which you can think of as a nonlinear filter for smoothing out curves. You can take your initial piecewise linear optimal path and evolve it under mean curvature flow for an arbitrarily short period of time $t>0$ and you will get a $C^\infty$ curve that is as close as you like (by making $t$ small) to the original curve, and necessarily has shorter length! This is also nice because the 3D version of mean curvature flow describes how a soap bubble moves into its equilibrium position.

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There are $C^{\infty}$ functions that approximate piecewise linear functions arbitrarily closely. For example, $$\lim_{n\rightarrow\infty}\frac{\log(\exp(n x)+1)}{n}=\frac{x+|x|}{2}$$ Given a piecewise linear interpolation of your set of points (which has minimum arc length), it is thus possible to construct a $C^{\infty}$ function that approximates this piecewise linear interpolation arbitrarily closely, with corresponding arc length arbitrarily close to the minimum.

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