TITLE: Convex curves with many inscribed triangles maximizing perimeter
QUESTION [10 upvotes]: A classical nice result  of Euclidean geometry states that the triangles maximizing the perimeter among all inscribed triangles of a given ellipse constitue a one-parameter family. Precisely, for each point on the ellipse there is exactly one such a maximizing triangle with vertex on that point, which can also be viewed as a period-3 bouncing trajectory.
I am wondering is this property  is characteristic of the ellipses:

Let $C$ be a compact convex subset of
  the plane such that for any point
  $P\in\partial C$ there is a triangle
  maximizing the perimeter among all
  inscribed triangles, with $P$ as a
  vertex. Is it true that $\partial C$ is
  necessarily an ellipse?

REPLY [2 votes]: Please see a relevant paper
Yu. Baryshnikov, V. Zharnitsky, Sub-Riemannian geometry and periodic orbits in classical billiards, Math. Res. Lett., 13 (2006), 587–598.
As far as I know, the following question is open: are there curves, rather than ellipses, that possess circles of p- and q-periodic billiard trajectories for different p and q.