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 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.