Horn's problem in PU$(n,1)$

The multiplicative Horn problem is the following question: given three conjugacy classes $\mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3$ in a Lie group $G$, do there exist elements $(A,B,C)\in\mathcal{C}_1\times\mathcal{C}_2\times\mathcal{C}_3$ such that $ABC=\operatorname{Id}$? In this paper, we study the multiplicative Horn problem restricted to the elliptic classes of the group $G={\rm PU}(n,1)$ for $n\geq 1$, which is the isometry group of the $n$-dimensional complex hyperbolic space. We show that the solution set of Horn's problem in PU$(n,1)$ is a finite union of convex polytopes in the space of elliptic conjugacy classes. We give a complete description of these polytopes when $n=2$.