Quasi-homogeneous domains and convex affine manifolds

In this article, we study convex affine domains which can cover a compact affine manifold. is purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least C1 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 Rn up to affine equivalence. Furthermore we prove that if a strictly convex quasi-homogeneous projective domain is Cα on an open subset of its boundary, then it is Cα everywhere. this fact and the properties of asymptotic cones we find all possible shapes for developing images of compact convex affine manifolds with dimension ⩽4.