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. My question then is: 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, where X(x) is the distance from a fixed point in the interior to the surface in the direction of x. Now let B ⊂ A be those solids having only one stable equilibrium point (local minimum of X, when the interior point is the centre of mass). 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.