Let $G$ be a compact matrix Lie group under the Killing form metric $\langle \xi, \eta \rangle\_g = -\frac{1}{2}\text{tr}((g^{-1}\xi)^T(g^{-1}\eta))$ for $g \in G$ and $\xi,\eta \in T\_gG$. Let $C \subset G$ be a geodesically convex set. Pick finitely many $g\_1,...,g\_N \in C$ and define $\Omega$ to be the smallest closed convex set containing those points. What does the boundary of $\Omega$ look like? Is it like the smallest geodesic polygon that fits those points, like in Euclidean space? Or is it something more complicated?



What does the boundary of convex hulls look like in matrix Lie groups?
I guess you wanted to say *smallest geodesic polytope* (not *polygon*).


It is unclear what is polytope in a the matrix Lie group, but it seems to require geodesic hypersurfaces.
They do not exist in most Riemannian manifold starting from dimension 3 and matrix Lie groups are not exceptional.


BTW, if you are interested in convex sets in general Riemannian manifolds,
then check our paper ["About every convex set..."](https://arxiv.org/abs/2103.15189).


