My question is similar to The mean of points on a unit n-sphere $S^n$.

I have a unit $n$-sphere $S^n$ and a set $P$ of points lying on its surface. I use geodesic distance metric $d(p,q)=\arccos(pq^T)$. Additionally I have a guarantee that all of my points are positive (unit vectors with all coordinates greater than or equal to 0).

The task is to find a centroid of those points, whose uniqueness is guaranteed because all points are positive.

In such case, does the centroid always coincide with the normalised arithmetic mean of the points? Formally \begin{gather*} \mu_\text{arithmetic} = \frac{1}{\lvert P\rvert}\sum_{p\in P}p \\ \mu_\text{geodesic}=\arg\min_c\sum_{p\in P}d(p,c)^2. \end{gather*} Question: do we have $$\mu_\text{geodesic}=\frac{\mu_\text{arithmetic}}{\lVert\mu_\text{arithmetic}\rVert}?$$