TITLE: surfaces of constant centro-affine curvature
QUESTION [9 upvotes]: It is well-known that every closed surface in $\mathbb R^3$ having constant Gauss curvature is a round sphere. Inspired by this question, 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.

REPLY [2 votes]: "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?"
YES, it's true ! An elegant short proof may be found in the following paper:  
Leichtweiss, Kurt On a problem of W. J. Firey in connection with the characterization of spheres. Festschrift for Hans Vogler on the occasion of his 60th birthday. Math. Pannon.  6  (1995),  no. 1, 67–75.