This question has been bugging me for awhile since some papers claim it is, and others do not. Suppose $M \subset \mathbb{R}^3$ is a sphere and sub manifold of 3D Euclidean space. Let $x$ be the North Pole, and suppose I have a geodesic ball $B$ centered at $x$ that is itself contained within the upper hemisphere. Is this set totally convex? My intuition says yes, because all geodesics whose endpoints are in $B$ are clearly contained in $B$. All geodesics are arcs of great circles, after all.
However, the textbook Convex Functions and Convex Optimization Methods" claimed that "in a sphere. any proper subset is not totally convex." Can someone point out my misunderstanding?