Conformal Barycenters in Quaternionic Hyperbolic Balls

Abstract

We extend the notion of conformal barycenter, recently introduced by Jačimović and Kalaj for the complex hyperbolic ball, to the quaternionic unit ball $\BH$. The quaternionic conformal barycenter of a measurable set $D$ with finite hyperbolic measure and finite first moment is defined as the unique point $c$ such that $\int_D Φ_c(q)\, \dLam(q) = \mathbf{0}$, where $Φ_c$ is the quaternionic Hua involution exchanging $0$ and $c$. Equivalently, it is the unique minimum of the energy functional $G(x) = \int_D \log\cosh^2\!\big(\frac12 d_H(x,y)\big)\, \dLam(y)$. We prove existence and uniqueness using the strict geodesic convexity of $G$, which is established by a direct computation along geodesics. The barycenter is invariant under the full isometry group $\mathrm{Sp}(n,1)$. We also treat finite point sets and provide explicit examples.