It is well-known that every closed surface in $\mathbb R^3$ having constant Gauss curvature is a round sphere. Inspired by [this question](https://mathoverflow.net/questions/61143), I'd like to ask whether a similar rigidity holds for centro-affine curvature.


More precisely, let $M\subset\mathbb R^3$ be a smooth closed convex surface (i.e., the boundary of a convex body) enclosing the origin. Its centro-affine curvature at a point $p\in M$ can be defined as
$$
K(p)\cdot\langle p,\nu(p)\rangle^{-4}
$$
where $K(p)$ is the Gauss curvature and $\nu(p)$ is the outer normal vector at $p$. (More generally, for a hypersurface in $\mathbb R^n$ it is $K(p)\cdot\langle p,\nu(p)\rangle^{-(n+1)}$.)


The nice thing about centro-affine curvature is that it is invariant under volume-preserving linear transformations. In particular, it is constant if $M$ is an ellipsoid centered at the origin (because such ellipsoids are equivalent to spheres). Is the converse true? In other words, is it true that every closed convex surface with constant centro-affine curvature is an ellipsoid?


**Remark.** For curves in $\mathbb R^2$, the condition that the centro-affine curvature is constant boils down to a second-order ODE whose solutions are ellipses and only ellipses. In $\mathbb R^3$, it is a PDE that seems to have many more solutions locally (just like in the case of constant Gauss curvature). So, if there is rigidity, it should be global only.


[EDIT] Another possible definition of centro-affine curvature is what the Legendre transform (the natural bijection between a convex body and its polar) does to the volume locally. I'm adding 'convexity' tag because some extremal properties of ellipsoids might be relevant here.



surfaces of constant centro-affine curvature
This is a classical question that was probably posed by Blaschke and partially solved by Jorgens and Calabi but I think was finally solved completely by Cheng and Yau in "Complete Affine Hypersurfaces. Part I. The Completeness of Affine Metrics" in CPAM 1986, if you assume sufficient regularity of the boundary. These bodies can be defined without any assumption on regularity except for convexity, but I am less sure about what is known in that case.


Description of centro-affine curvature: Given a convex body with the origin in the interior, there is a uniquely defined convex function $\rho$ homogeneous of degree $2$ that is equal to $1$ along the boundary of the body. If the body is origin-symmetric, then it is the unit ball of a Banach norm and $\rho$ is just the norm squared. The Hessian of $\rho$ is homogeneous of degree $0$. The determinant of the Hessian is the centro-affine curvature.


