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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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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... –  Joseph O'Rourke Jul 15 '10 at 22:30
    
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. –  Robin Saunders Jul 15 '10 at 22:47
    
@Robin: I see! Thanks for explaining. –  Joseph O'Rourke Jul 15 '10 at 22:56
    
Can you perhaps rewrite the question to make is complete and precise? I am note sure I understand "What about...?" type of question. –  Igor Pak Jul 15 '10 at 23:05
    
The precise question itself is stated further down; the "what if...?" was just used to introduce it. Sorry if that was a bit confusing. Ignoring the introductory background paragraph though, the rest of the question should be self-contained. Does it make sense, or is there something I could explain more clearly? –  Robin Saunders Jul 15 '10 at 23:32
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