My question is similar to 

https://mathoverflow.net/questions/231501/the-mean-of-points-on-a-unit-n-sphere-sn

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 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 normalised arithmetic mean of the points? Formally
$$\mu_{arithmetic} = \frac{1}{|P|}\sum_{p\in P}p$$
$$\mu_{geodesic}=\arg\min_c\sum_{p\in P}d(p,c)^2$$
Question:
$$\mu_{geodesic}=\frac{\mu_{arithmetic}}{||\mu_{arithmetic}||}$$