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This made the popular news a few years ago. Summary: it's the first homogeneous, convex solid to be found with only one stable and one unstable mechanical equilibrium when resting on a flat surface. I found the notion pretty interesting and visited the discoverers' website, where they describe attempts to discover a smooth version which technically succeeded, but they were unable to build a physical model due to extremely high sensitivity to imperfections. The piecewise smooth version they ended up with is still very sensitive to such imperfections, but less so. So then I wondered: what if we no longer care about the number of unstable equilibria?

Let A be the set of smooth, bounded, convex solids (assumed to be homogeneous). An element of A can be thought of as an immersion X : S2 → R3; we can define oX to be the centre of mass of the solid interior. Now let B ⊂ A be those solids having only one stable equilibrium point (local minimum of |X-oX|). Then we can define:

d(x,Y) = infy∈S^2|X(x)-Y(y)|

r(X,Y) = supx∈S^2(d(x,Y)))

and finally rmin(X) = infY∈A\B(r(X,Y)), the size of the "safety margin" around X ∈ B. Then the question is: which unit-volume X ∈ B maximizes rmin(X), i.e. is the least sensitive to imperfections?

With regard to the tagging: I suppose this is really an exercise in calculus of variations, but this was the closest I could get using the arXiv tags.

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  • $\begingroup$ How about a nearly flat cone with a flat base (rounded appropriately near edges to make it smooth)? Then the wide base will be a safe stable equilibrium point, and that seems to be the only stable position. Perhaps I am misinterpreting your question... $\endgroup$ Jul 15, 2010 at 22:30
  • $\begingroup$ Technically you'll have a circle of "stable equilibria" near the apex of the cone. I know they're not truly stable, but informally they are "more like stable equilibria than anything else". More formally, they are still local minima of |X|. If placed on one of these points, the cone will roll around in circles, eventually coming to rest (due to friction) on a point other than the base. You'd also have to be careful rounding off edges in general, to avoid introducing extra local minima there. $\endgroup$ Jul 15, 2010 at 22:47
  • $\begingroup$ @Robin: I see! Thanks for explaining. $\endgroup$ Jul 15, 2010 at 22:56
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    $\begingroup$ Well, it's pointed out on the creators' website (www.gomboc.eu) and also in the Wikipedia article that any plane curve must have at least two stable equilibria (as well as at least two unstable equilibria), as a consequence of the four-vertex theorem. So a surface of revolution wouldn't work even with my loosened criteria. $\endgroup$ Jul 16, 2010 at 21:14
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    $\begingroup$ @Robin: a three-dimensional surface of revolution isn't the same as a constant-mass plane curve, so it's not clear to me that this theorem directly applies. $\endgroup$
    – Peter Shor
    Jul 17, 2010 at 18:25

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