Quasi-homogeneous domains and convex affine manifolds

In this article, we study convex affine domains which can cover a compact affine manifold. For this purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least $C^1$ boundary and it is an ellipsoid if its boundary is twice differentiable. And then we show that an n-dimensional paraboloid is the only strictly convex quasi-homogeneous affine domain in $\mathbb R^n$ up to affine equivalence. Furthermore we prove that if a strictly convex quasi-homogeneous projective domain is $C^{\alpha}$ on an open subset of its boundary, then it is $C^{\alpha}$ everywhere. Using this fact and the properties of asymptotic cones we find all possible shapes for developing images of compact convex affine manifolds with dimension $\leq 4$.