<a href="http://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c">This</a> 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<sup>2</sup> → R<sup>3</sup>; we can define o<sub>X</sub> 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<sub>X</sub>|). Then we can define: d(<b>x</b>,Y) = inf<sub>y∈S^2</sub>|X(<b>x</b>)-Y(<b>y</b>)| r(X,Y) = sup<sub><b>x</b>∈S^2</sub>(d(<b>x</b>,Y))) and finally r<sub>min</sub>(X) = inf<sub>Y∈A\B</sub>(r(X,Y)), the size of the "safety margin" around X ∈ B. Then the question is: which unit-volume X ∈ B maximizes r<sub>min</sub>(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.