The classical sources on projectively equivalent connections are Levi-Civita: Sulle trasformazioni delle equazioni dinamiche. Ann. di Mat., serie 2a. 24, 55–300 (1896). English transl. in Regular and Chaotic Dynamics 14, 580–614 (2009) and Thomas, T. Y.: On the projective and equi-projective geo metries of paths. Proc. Natl. Acad. Sci. USA 11, 199–203 (1925) In recent time projectively equivalent connections were actively studied in the framework of parabolic geometries; see the survey M. Eastwood, Notes on projective differential geometry, arXiv:0806.3998 Everything IMHO

Juan Carlos, if all the curves of constant geodesic curvature of a metric are circles, then the geodesics are circles as well which implies (in dimension 2) that the metric has constant curvature by the classical result of Segre, see the link below ams.org/mathscinet/search/…

Guiseppe, I do not know what to tell you. You have asked whether you are missing something -- no, you are missing nothing and your explanation is more precise than mine one.One can make mine explanation also mathematically precise (because your can build darboux coordinate system such that the differential of the first two coordinates coincide with $dC_1$ and $dC_2$ provided the poisson bracket of $C_1$ and $C_2$ is zero at a point.) This observation will be enough to justify my answer, and you do more or less the same in your answer.

Juan Carlos, I do not insist on invariance or equivariance, I insist that I am not confusing something, at least now. I did have some problems from the very beginning, and it seems that youe are repeating my way and make the same mistaked. If you want we can stop this discussion, of cause; if now, please explain me what you understand under ``such'' in your last comment.

To alvarezpaiva: If you look on the initial question of Deane, you can see that it is indeed does not change if we scale of the convex body -- simply look on the formula. It is because it does not coincide with the Lutwak-Yang-Zhang construction, the difference is a factor which makes the Lutwak-Yang-Zhang construction canonic. I do not think that I confusing "invariant" and "equivariant", especially because in all editions of my answer I supported the words by formulas

Dear Guiseppe, you are right. To construct the Darboux coordinates we indeed need that the bracket of $C_1$ and $C_2$ vanish in a small neighborhood. Would the bracket of $C_1$ and $C_2$ vanish at the points of $\mathcal{N}$ only, one can find Poisson-commute functions $\tilde C_1$ and $\tilde C_2$ whose differentials at the point we consider coincides with that of $C_1$ and $C_2$, and construct Darboux coordinates starting from $\tilde C_1$ and $\tilde C_2$, which would be sufficient.

No, non of my current understanding contradicts your comments and would I read them more carefully and spend more time thinking about them I could probably save few hours of calculations. Would you say directly that the construction is not canonic it would be easier for me to understand the matter. Anyway it is a nice construction and it was a fun to think about it :-)

Dear Deane, I put the link to the picture related to the counterexample. I hope it would help you when you have time to check whether it is indeed a counterexample. I did not understand your last two comments, sorry. Though you do use the inner product of $R^2$, you claim that the construction is invariant with respect to the group $Sl_2$. In my counterexample, the dependence of the ellipsoid on the linear transformations is not as you claim. It does behave as you claim under the re-scalings, which was clear from the very beginning, and rescaling of the inner product would not help me.