  In a Riemannian manifold a regular convex domain is said to be
$\lambda$-convex if its normal curvature at each point is greater than or equal
to $\lambda$. In a Hadamard manifold, the asymptotic behaviour of the quotient
$\vol(\Omega(t))/\vol(\partial\Omega(t))$ for a family of $\lambda$-convex
domains $\Omega(t)$ expanding over the whole space has been studied and general
bounds for this quotient are known. In this paper we improve this general
result in the complex hyperbolic space $\CH$, a Hadamard manifold with constant
holomorphic curvature equal to $-4k^2$. Furthermore, we give some specific
properties of convex domains in $\CH$ and we prove that $\lambda$-convex
domains of arbitrary radius exists if $\lambda\leq k$.
