# What does the boundary of convex hulls look like in matrix Lie groups?

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 are $M$ and $C$? (Did you mean $M \subset G$ instead of $G \subset M$?) May 3, 2022 at 17:08
• Maybe meant $C\subseteq G$. May 3, 2022 at 17:59
• @LSpice Apologies. I meant $C \subset G$. I edited my question. May 3, 2022 at 18:33
• You did not specify the metric on $G$. Are you assuming that $G$ is compact and endowing it with the Killing metric ? In that case, $G$ is not uniquely geodesic, so convex hulls are not a very robust notion (and it is not even clearly defined globally). May 3, 2022 at 20:14
• Perhaps a first case to look at is $SU(2) = S^3$. How do you define convex hulls in a sphere ? May 3, 2022 at 20:21

• @SpencerKraisler It does: Given a positive integer $k$, we say that a property $P$ holds for $C^k$-generic Riemannian metric $g$ on a manifold $M$ if the property $P$ holds for a dense G-delta set (that is, a countable intersection of open subsets) of metric tensors in the $C^k$-topology. Feb 2, 2023 at 12:50
• @SpencerKraisler, You can do it, but I what for? For example, let us assume that "property" is something that can be expressed in a finite number of words; so there are only countable set of properties. Then say that manifold $M$ is generic if any generic property holds for $M$. But in this case any manifold that can be completely described in finite number of words is not generic, so it is not possible to describe it explicitly despite most of manifolds are generic. Feb 2, 2023 at 19:35