  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 $\leq 4$.
