  In this paper, we show that any convex affine domain with a nonempty limit
sets on the boundary under the action of the identity component of the
automorphism group cannot cover a compact affine manifold with a parallel
volume, which is a positive answer to the Markus conjecture for convex case.
Consequently, we show that the Markus conjecture is true for convex affine
manifolds of dimension $\leq 5$.
