Finding the shortest curve that is at distance $\epsilon$ of every point of a surface

Question

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.

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? — Anton Petrunin, Jan 15 '17 at 20:47
@Wojowu "intrinsication" = "passing to the induced intrinsic metric" — Anton Petrunin, Jan 15 '17 at 21:36
@AntonPetrunin, I'm not sure I understand your comment, what do you mean by "compactness of its intrisication"? I don't see what "more general" condition you are refering to? — LCO, Jan 16 '17 at 09:03
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." — Joseph O'Rourke, Jan 16 '17 at 13:51
@LCO "intrinsication" is a metric space, if it is compact then yes, there is an optimal curve (likely not smooth). — Anton Petrunin, Jan 16 '17 at 18:57

score 1 · Answer 1 · answered Jan 16 '17 at 02:02

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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.

answered Jan 16 '17 at 02:02

Victor Protsak

14,347

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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. – LCO Jan 16 '17 at 09:10
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. – Victor Protsak Jan 16 '17 at 16:42

Finding the shortest curve that is at distance $\epsilon$ of every point of a surface

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