Convex curves in gear (Q1071316)

The author considers the two upper and lower radii of curvature \(\rho_ s^{\pm}\) and \(\rho_ i^{\pm}\) at an arbitrary point of a convex curve and obtains that for most convex curves (i.e. all except those in a set of first Baire-category) \(\rho_ s^{\pm}=\infty\) and \(\rho_ i^{\pm}=0\). The author uses this result to interpret two convex curves which touch at a common point in a special way, as convex curves in gear. He shows for these configurations that in a certain sense (which is precised in the paper) most pairs of convex curves are in gear.