Let $G$ be a matrix Lie group under a left-invariant metric. 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?

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?

Let $G$ be a matrix Lie group under a left-invariant metric. Let $G \subset M$ 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?

Let $G$ be a matrix Lie group under a left-invariant metric. 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?