Is there a nice formula/method to find the convexity radius of a matrix Lie group (the manifold can be noncompact) ?
Edited based on comments:
Definition : Convexity Radius (Berger - Panoramic View of Riemannian Geometry) The convexity radius of a Riemannian manifold $M$ is the infimum of positive numbers $r$ such that the metric open ball $B(m,r)$ is convex for every $m ∈ M$.