Let $G$ be a semisimple Lie group of rank one and let $\Gamma$ be a convex-cocompact, Zariski dense subgroup. Let $X=G/K$ denote the symmetric space and $\partial X$ its visibility boundary. Let $\Lambda(\Gamma)\subset\partial X$ be the limit set and let $C(\Gamma)\subset X$ be the convex hull of the limit set (intersected with $X$).

My question is about the shape that $C(\Gamma)$ can take. Does it have interior points? Is it the closure of its interior points? Is it a smooth Riemmannian manifold with boundary (or corners)? If not, is this true if $\Gamma$ is a subgroup of an arithmetic group?