I have stumbled across the following meta-question while working on a research problem.
I have a compact, convex set, $\mathcal{K}$. Be Krein-Milman, any element in $\mathcal{K}$ can be written as convex combination of its extreme points. I am able to characterize the extreme points of the set. Moreover, I also have that a subset of the extreme points, $\mathcal{F}$, form a compact, connected Lie Group Denote its Lie algebra by $\mathfrak{f}$. For context, see here:
https://www.sciencedirect.com/science/article/pii/S039304401500042X
Every element in $\mathcal{F}$ can be written as $\exp(f)$ for $f \in \mathfrak{f}$. Hence, every element of $\text{Conv}( \mathcal{F} ) \subset \mathcal{K}$ can be written as a convex combination of such exponentials.
Is the best one can do? Ideally, I am interested in finding Lie algebra generators of elements in $\text{Conv}( \mathcal{F} )$ ( or better yet $\mathcal{K}$).
Can someone suggest approaches, references for this problem?