Let $\gamma\_1,\gamma\_2: \mathbb{S}^1 \to \mathbb{R}^2$ be two smooth, closed, convex curves that their [(special)affine curvature](http://en.wikipedia.org/wiki/Affine_curvature), $\mu\_1,\mu\_2$ are equal, that is $\mu\_1(\theta)=\mu\_2(\theta)$, for any $\theta\in \mathbb{S}^1$. Are$\gamma\_1,\gamma\_2$ related to each other by an affine transformation and reparameterization? Are these two curves, $\gamma\_1(\mathbb{S}^1 ),\gamma\_2(\mathbb{S}^1) $, are related by an affine transformation?



Uniqueness affine curvature
Note that closed curves made by two arcs of ellipses of the given areas, say $a$ and $b$ may be not related to each other by an affine transformation.
They have constant affine curvatures on two arcs, so for suitable parametrization the curvatures are the same. 


The example is $C^{1,1}$-smooth, but it can be $C^\infty$-smoothed.


