Strongly convex set

Definition

Definition with symbols

A subset $C$ of a Riemannian mannifold $M$ is said to be strongly convex if given any two points $p,q\in C$, there is a unique normal minimal geodesic joining $p$ with $q$, and moreover, this geodesic lies completely within $C$.

Facts

Given any point, a metric ball of sufficiently small radius about that point is strongly convex.