Dirichlet polyhedra for cyclic groups in complex hyperbolic space

We prove that the Dirichlet fundamental polyhedron for a cyclic group generated by a unipotent or hyperbolic element $\gamma$ acting on complex hyperbolic $n$-space centered at an arbitrary point $w$ is bounded by the two hypersurfaces equidistant from the pairs $w,\gamma w$ and $w,{\gamma ^{ - 1}}w$ respectively. The proof relies on a convexity property of the distance to an isometric flow containing $\gamma$.