The mean of positive points on a unit $n$-sphere $S^n$

Question

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 $$\mu_\text{geodesic}=\frac{\mu_\text{arithmetic}}{\lVert\mu_\text{arithmetic}\rVert}?$$

Answer 1

The answer is no. E.g., let $P$ be the set $\{1,e^{it},e^{i3t}\}$ of points on the unit circle in $\mathbb C=\mathbb R^2$, where $t$ is a small positive real number. Then the geodesic mean of $P$ is $$e^{i(4/3)t}=1+\frac{4 i t}{3}-\frac{8 t^2}{9}-\frac{32 i t^3}{81}+O\left(t^4\right),$$ whereas the arithmetic mean of $P$ is $$\frac{1+e^{i t}+e^{3 i t}}{\left| 1+e^{i t}+e^{3 i t}\right| } =1+\frac{4 i t}{3}-\frac{8 t^2}{9}-\frac{14 i t^3}{27}+O\left(t^4\right),$$ so that the geodesic mean of $P$ differs from the arithmetic mean.