My grad school days are 30 years behind me, and to my chagrin proving what seems a simple observation draws upon skills which are too badly atrophied to be useful. The inspiration is a wine bottle. Why do wine bottles have "punts" - i.e. indentations on the bottom? Obviously so that you can stand the bottle on its end and it won't tip over. Can we prove that is always true?
Consider two continuously differentiable surfaces, $A, B$ in $\Bbb R_3$, $A$ convex; continuously differentiable closed curves, $a, b$ on each; and a mapping, $\Gamma:A\rightarrow B $, which maps $a \rightarrow b$, and $int(a) \rightarrow int(b)$, $ext(a) \rightarrow ext(b)$, such that the mapping preserves curvature at every point in the exterior, and inverts the curvature within in interior.
Prove $\Gamma$ is a plane curve.
For the special case of $A$ being a sphere, $B$ is a sphere with a dimple, and the proof is as simple as noting that the intersection of two spheres is a circle. Sadly, that isn't much of a start on the general case.
