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I was recently explaining to a friend what the convex hull of a set of points is using the analogy of an elastic band around a set of nails hammered into a board. I was about to say that we can generalize this to three-dimensions by replacing the elastic band with shrink wrap, but this is false! For example, if you shrink wrap a dumbell you will get an hourglass shape.

Is there is an appropriate physical process that really does give the convex hull? If the answer is no, is there some reason why we shouldn't expect physically reasonable minimization problems to give convex hulls?

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    If the shrink wrap is tight enough shouldn't everything work out?2012-06-04
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    Here is my reasoning: in the limit as the dumbell gets very long, surface area is minimized when the shrink wrap at the middle of the dumbell stays closer to the bar.2012-06-04
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    I agree that the cylinder is not a minimal surface, but would warn against the assumption that the process of shrink-wrapping minimizes surface area. It is more appropriate to think that it minimizes some stored-energy functional (which depends on the material, and which we don't really know).2012-06-04
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    Gift wrapping using an inextensible material like paper might work. You can make a sheet of paper conform to a surface of positive Gaussian curvature by folding it, but negative Gaussian curvature will tear the paper. My intuition suggests that the convex hull is the minimal surface whose Gaussian curvature is nonnegative everywhere, but I don't know if this is true.2012-06-04
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    My conjecture seemed like an interesting enough question that I didn't know how to answer, so I've [asked it as a separate question](http://math.stackexchange.com/q/153971/856).2012-06-04
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    Yes, this looks like the kind of answer I was looking for.2012-06-06

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