Invariance of the area of ovaloids

Abstract

Consider a two dimensional smooth convex body with a \linebreak marked point on the boundary of it, sitting tangentially at the marked point over a base curve in $\EEE^2$, $\HHH^2$ or $\SSS^2$ and the image of this body by the reflection with respect to the tangent line of the base curve at the marked point. When we roll these two bodies simultaneously along the base curve, the trajectories of the marked point bound a closed region. We show that the area of the closed region is independent of the shape of the base curve if the base curve is not highly curved with respect to the boundary curve of the convex body.