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.