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 : S^{2} → R^{3}; we can define o_{X} 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-o_{X}|). Then we can define:

d(**x**,Y) = inf_{y∈S^2}|X(**x**)-Y(**y**)|

r(X,Y) = sup_{x∈S^2}(d(**x**,Y)))

and finally r_{min}(X) = inf_{Y∈A\B}(r(X,Y)), the size of the "safety margin" around X ∈ B. Then the question is: which unit-volume X ∈ B maximizes r_{min}(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.