Consider a planar convex region $C$. Let us define a mapping of a point $P$ on $C$ to that point on C that is farthest from $P$. Obviously, if from an initial position of $P$, we do this mapping repeatedly, $P$ traces out a trajectory.
For example, if $C$ is a rectangle, from any point $P$ on it, the mapping takes $P$ to a corner of $C$ and further applications of the mapping merely causes $P$ to keep switching between two ends of a diagonal.
Question: Given any positive integer $n$, can one find some $C$ and some initial $P$ on it such that the trajectory of $P$ when the mapping is repeated $n$ times has all the $n$ points different? If for a $P$, there are many different farthest points to it, we are free to choose any one from them as the next point in the trajectory and proceed.
Observations: We checked numerically the case when $C$ is an ellipse and $P$ is initially a random point on the boundary. If the eccentricity of $C$ is low, we find that the number of times the mapping can be done before the position of $P$ coincides with its position in an earlier iteration is quite high (we could go up to 40) but seems always finite. It also seems that for any ellipse, the initial position of $P$ that maximizes the number of iterations before position of $P$ repeats is an end of the minor axis of $C$. If $C$ has high eccentricity, $P$ repeats after very few iterations. It is not clear with the precision that we could manage whether $P$ really hits a repetition with finitely many iterations or only tends rapidly to a repetition but actually needs infinitely many iterations to get there.
Further questions: Going to 3D, one can have multiple variants of this question by (1) considering $C$ as a solid 3D convex region or (2) the surface of such a region with the farthest point to any $P$ being defined either in terms of geodesics or in terms of paths which lie entirely on a plane (these two definitions also feature in 'Uniformity' of surfaces of 3D convex solids). One can also ask about metrics other than Euclidean.
