# Math Help - Greatest convex geodesic ball in Grassmannian

1. ## Greatest convex geodesic ball in Grassmannian

Hello,

I´ve got a question concerning convex balls:

Let 0<k<n and G(n,k) denote the Grassmannian of (non-oriented) k-planes in \IR^n. If P is a k-plane in G(n,k), what is the greatest radius r>0, such that the geodesic ball
B_r(P) is convex?

Thank you for helping me!
Regards,
Kevin

2. Consider all radial geodesics $g$ eminating from $P$. Then, the infimum of injectivity radii $m$ over the set $\{g\}$ is attained, as $G(n,k)$ is compact. Maybe you will be required to additionaly show the ball $B(m)$ is convex,
a fact I find reasonable... but can't produce a formal proof at the moment. :P

ps. disregard that. it's fairly simple