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 $m ∈ M$.