An algorithm for approximating the metric projection onto a superelliptic disk
We study conditions under which the metric projection onto a strictly convex body can be approximated by the projections onto polygonal approximations (in the Hausdorff-Pompeiu sense) of the strictly convex body itself. We provide a necessary condition by proving that the union of all the vertices of the polygonal approximations must be dense in the boundary of that set. To showcase our result, given a superellipse (Lamé curve) of order p>1, we prove that a sequence of polygons obtained by radially projecting the torsion subgroup of the unit circle onto the superellipse, converges to the superellipse, in the Hausdorff-Pompeiu metric. Moreover, this leads to a globally convergent new algorithm useful in approximating the metric projection onto a superelliptic disk (the convex hull of a superellipse).