Convexity radius

Definition

Definition with symbols

Let $p$ be a point in a Riemannian manifold $(M,g)$. Then, the convexity radius at $p$ is defined as the supremum of all radii $r$ for which the following two conditions hold:

• Any metric ball contained in the metric ball $B_r(p)$ is strongly convex
• Any geodesic segment contained in $B_r(p)$ is a minimal geodesic joining its endpoints