Robert, I read his answer differently, I assumed that the author asks whether the 4th power of the distance function in an arbitrary finsler metric is smooth. If your version of his answer is true, my answer is indeed irrelevant. Let us wait the comment or the explanation of the author

Juan Carlos, what is your definition of the legendre ellipsoid?

3 continued: I think you now see the phenomena: In order to make the set of metrics smaller, the orbit should be dense and should return close to a point with different differentials. I do not think that one can construct a two-dimensional example with one-dimensional set of metrics but I think I know an example, though complicated one, in dimension 4

3. Let me give an example, again on the torus, such that the set of the metrics is two-dimensional. It is similar to my example 2, but instead of shifting w.r.t vector $(\alpha_1, \alpha_2)$ take the sliding symmetry with respect to this vector. Since the square of the sliding symmetry is translation, the metric must have constant components in our coordinate system. The condition that it should be preserved by the sliding symmetry is an additional restriction on the metric that kills one from thee freedoms in changing the metric.

2. Take any orbit and its closure. Denote by $d$ the distance to the orbit (in a background metric for which your mapping is an isometry). The function $d$ is preserved by isometries. Choose any positive function $f$ and conformally change the metric by multiplying it by $f(d)$. You will obtain a metric which is preserved by isometries, and you have a functional freedom, namely a choice of a function $f$. If the function $f$ is such that it is nonconstant only for small values, the resulting metric will be a smooth one.

1. The ratio should be irrational because otherwise the orbit of any point will be discrete and therefore not dense. Like in my first example.

May be it would help if you give other definitions you are working with: for example you said ``triangle''. I assume that the sides are geodesics. Are they minimal geodesics or simply geodesics?

I misunderstood your question in which actually you did not specify that your metric is of constand curvature. Indeed, you are right, in my example I changed the geometry

just take a hyperbolic surface admitting a self-isometry, take a small triange ABC on it and its image A'B'C' w.r.t. the self-isometry, and slightly change the metric in a small neightborhood inside the first triangle. You would not change the edges but there will be no isometry that carries the first to the second, and their interiours have different volume.

It is a trick, it does not show the geometry behind, at least for me. I thought about it when I gave an advanced course of lectures on classical mechanics, and decided to do a more conventional approach. But I agree that it is a nice trick