Conformal Barycenters in Quaternionic Hyperbolic Balls

We extend the notion of conformal barycenter, recently introduced by Jačimović and Kalaj for the complex hyperbolic ball, to the quaternionic unit ball $\text{BH}$. 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 ${\int }_D{\mathrm{\Phi }}_c(q)\text{dLam}(q)=\mathbf{0}$, where ${\mathrm{\Phi }}_c$ is the quaternionic Hua involution exchanging $0$ and $c$. Equivalently, it is the unique minimum of the energy functional $G(x)={\int }_D\log {\cosh }^2\text{\hspace{0.17em}⁣}(\frac{1}{2}{d}_H(x,y))\text{dLam}(y)$. 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 $\mathrm{S}\mathrm{p}(n,1)$. We also treat finite point sets and provide explicit examples.