Convex curves with many inscribed triangles maximizing perimeter 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?

 A: This 1988 paper of Innami gives a construction of a convex curve, all of whose points are vertices of billiard triangles and therefore also maximal-perimeter inscribed triangles. In Innami's construction, all triangles are isosceles.
A: 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. 
A: N.B.  This is an edit of my original post, confirming the guess that I made originally.
The answer is no, i.e., such curves are not forced to be ellipses.
Here is a sketch of the argument.  (The details will take a while to type in, and that will have to wait.  Also, I'm not sure that there will be that many people interested in seeing the details, so I might not put them in unless I get a request.)
Such a curve $C$, if it is strictly convex (which I am assuming from now on) defines a closed curve in the $5$-dimensional space $T$ of triangles in the plane with fixed perimeter $P>0$.  (Note that $T$ is an $S_3$-quotient of a hypersurface in the space of triples of non-collinear points in the plane.)  These triangles of perimeter $P$ are so-called 'billiard triangles', i.e., the angle of incidence and reflection of the sides of each vertex of the triangle with the curve $C$ are equal. 
This means that such a curve in $T$ is tangent to a certain rank $3$ plane field $D\subset TM$, and, conversely, a closed embedded curve in $T$ that is tangent everywhere to this plane field $D$ represents a closed curve of triangles of perimeter $P$ moving in such a way that the velocity of each vertex is perpendicular to the triangle's angle bisector at that vertex.  Thus, the curves we want to study are solutions of an underdetermined system of ODE.  
The $D$-integral curves represented by the ellipses (which do give solutions, by Chasles' Theorem) are 'regular' in the sense of control theory (this is what I had to check by looking at the structure equations of $D$), so that one can make an arbitrary functions' worth of perturbations of any such closed $D$-integral curve (say, one given by an ellipse) to get other nearby $D$-integral curves.  Thus, there will be lots of such closed convex curves, near ellipses but not ellipses, that have a circle of inscribed triangles of maximum perimeter.  
It might be interesting to construct some explicit closed integral curves that don't come from ellipses, but I don't see any easy way to do that right now. 
Remark:  The plane field $D$ is interesting.  It is bracket-generating and does not contain an integrable $2$-plane field, so that it belongs to the type studied by Élie Cartan in his famous "Five Variables" paper of 1910.  It is not 'flat' in Cartan's sense, i.e., the group of symmetries of the plane field is not of dimension $14$, but rather of dimension $3$ (very small).  Still, the general theory says that the $D$-integral curves near the ellipses (in a suitable sense) are regular curves, so that they are freely deformable.
A: Dear Colleagues,
the question that was originally posed is slightly different:
Let C be a compact convex subset of the plane such that for any point P∈∂C there is a triangle maximizing the perimeter among all inscribed triangles, with P as a vertex. If the perimeter of maximizing triangle at P∈∂C  is CONSTANT, independent of P, is it true that ∂C is necessarily an ellipse?
Sorry for misunderstanding
Vladimir Georgiev
A: @Robert: ok, from your construction it seems that the answer in general is negative, but possible positive direction is to find the minimal regularity of integral curves that could "imply" the uniqueness, i.e. the boundary of the convex set is ellipse. Or it is possible to find closed smooth integral curves different from ellipse, satisfying the property to
have a circle of inscribed triangles of maximum perimeter. Probably real analiticity of the boundary can be helpful? 
