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Let $M$ be a compact connected (smooth) surface (possibly with boundary) in $\mathbb{R}^3$ and $\epsilon>0$ a constant.

Is there (and if there's not, what conditions on ($M$, $\epsilon$) should we add to have such existence) a (smooth) curve $\gamma$ on $M$ with minimal length such that for any point $x\in M$, there is a point $\gamma(t)$ on the curve such that $d(\gamma(t),x)\leq \epsilon$.

I am also interested, in the case of positive results of existence, on the method to obtain explicitly such minimal length curves.

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  • $\begingroup$ For any metric space, compactness of its intrinsication is sufficient. For sure one may get a more general condition, but why do you need it? $\endgroup$ – Anton Petrunin Jan 15 '17 at 20:47
  • $\begingroup$ @AntonPetrunin What is an "intrinsication"? $\endgroup$ – Wojowu Jan 15 '17 at 21:13
  • $\begingroup$ @Wojowu "intrinsication" = "passing to the induced intrinsic metric" $\endgroup$ – Anton Petrunin Jan 15 '17 at 21:36
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    $\begingroup$ Related: "Optimal inspection path on a sphere." That question asked for the curve when $M$ is a sphere. See, in particular, the reference to von der Mosel & Gerlach's work on "sphere-filling ropes." $\endgroup$ – Joseph O'Rourke Jan 16 '17 at 13:51
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    $\begingroup$ @LCO "intrinsication" is a metric space, if it is compact then yes, there is an optimal curve (likely not smooth). $\endgroup$ – Anton Petrunin Jan 16 '17 at 18:57
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If by "curve" you mean a continuous image of $[0,1]$ then the answer is almost certainly negative, even for flat surfaces, because the optimal subset need not be connected. For example, consider the union of two overlapping unit disks and $\epsilon=1$ (think of two rubber disks "stuck together" along a small part of their boundary). Then the optimal set is the two-point set consisting of their centers. If you perturb this configuration (and $\epsilon$) slightly, potential optimal sets would remain disconnected.

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    $\begingroup$ Thank you for you answer, but I don't see how this answers (negatively) the question. By curve, i do mean continuous (and even smooth). I don't see how the fact that there is an optimal set that is not a curve, imply that there is no curve that goes near every point at distance $\leq \epsilon$ , and such that any other curve satisfying this, will be longer. $\endgroup$ – LCO Jan 16 '17 at 9:10
  • $\begingroup$ My point was that you need to be much more careful with formulating your problem. The basic phenomenon (already for a unit disk and $\epsilon $ close to and less than 1) is that various optimal sets tend to be disconnected. Depending on the type of $\gamma $ you stipulate (e.g. Peano continuum or unicursal graph), possible solutions, if they exist, can vary greatly. I agree that, logically speaking, none of this precludes the pure existence in a specific class of $\gamma $; as for explicit construction, I am very pessimistic for the reasons outlined. Best wishes, Victor. $\endgroup$ – Victor Protsak Jan 16 '17 at 16:42

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