In Euclidean Space, given some points, we can find the smallest enclosing ball which contains all the points. And the center is known as the Chebyshev Center. But now I encounter a problem related to rotation and I would like to find the smallest enclosing ball on the manifold $SO_3$, using the metric of geodesic distance (1):$$ d(R, Q)=\operatorname{acos}\left(\frac{\operatorname{tr}\left(R^T Q\right)-1}{2}\right) $$ To make this problem make sense, we may consider the points contained in a big ball with radius $\pi$.
I'm thinking about this question for quite a while, but I can't find much progress. Now I can solve this problem with some iterative algorithms, but due to this problem's simple form, I'm wondering if there is an explicit solution. I'm wondering if I can use something like stereographic projection to simply the problem.
Or, if I only consider the circumstances of 3 or 4 points on the $SO_3$, can I hope to get an explicit solution to this Chebyshev center problem?(To verify my iterative method). Any hints and help will be appreciated, thanks a lot in advance!
Edit 2023/6/3:
After some searchings, I find that the geodesic distance above is the bi-invariant metric defined on $SO_3$ and the angular distance in $S^3$ is also bi-invariant. The bi-invariant metric is unique up to scaling if the Lie group is compact and Lie algebra is simple. We can conclude that both these two bi-invariant metrics are unique up to scaling.
Now if I consider a diffeomorphism $F: SO_3 \to RP^3$ and locally the $RP^3$ looks the same as $S^3$, maybe we can consider the homomorphism: $\rho: SO_3 \to S^3$ and somehow prove that after the homomorphism, the new metric is still bi-invariant (I'm not sure if I'm right). If this holds, then maybe I can just study about the problem in $S^3$ ? But I'm still not sure if I'm right and how to interprete these in a precise way.
Also I think my previous thought of using the stereographic projection to simply the problem is not valid, for the smallest ball in $R^3$ after stereographic projection isn't the smallest on the $S^3$. So I'm still wondering how to tackle this.
The key point is that there is a 2-to-1 mapping $S^3\to SO(3)$ where $S^3$ is viewed as the set of unit quaternions (see this wiki page for more on this map). This map preserves the standard geodesic distance (at least up to a constant scale factor depending on your convention). Using this covering map, we can associate each rotation $R\in SO(3)$ with a pair of antipodal points $\pm q\in S^3$. Given a set of rotations $R_1,\cdots,R_n$, there is a corresponding set of pairs $\pm q_1,\cdots,\pm q_n$, and the problem is equivalent to finding the smallest geodesic ball in $S^3$ which contains at least one member of each pair.
Since every geodesic ball in $S^3$ is the restriction of a geodesic ball in $\mathbb{R}^4$, you can find the minimal ball in $S^3$ by finding the minimal ball in $\mathbb{R}^4$ and computing its restriction to $S^3$