Conformal and holomorphic barycenters in hyperbolic balls

We introduce the notions of $\mathit{\text{conformal}}\mathit{\text{barycenter}}$ and $\mathit{\text{holomorphic}}\mathit{\text{barycenter}}$ of a measurable set $D$ in the hyperbolic ball. The two barycenters coincide in the disk, but they differ in multidimensional balls ${\mathbb{C}}^m\cong {\mathbb{R}}^{2m}$. These notions are counterparts of barycenters of measures on spheres, introduced by Douady and Earle in 1986.