I've read in several places that the convex hull of a subset of the sphere is not the whole sphere if and only if the subset is contained in an open hemisphere, but I'm not sure how to prove it. Does anyone know a proof?
By the sphere, I mean the unit sphere in any finite dimension, endowed with its intrinsic metric. By convex hull of a subset, I mean the smallest set containing that subset that is closed under taking shortest paths between points.
CLARIFICATION
Everything in this question is occurring inside the unit sphere, i.e., in the metric space $\mathbb{S}^n$, endowed with its intrinsic metric measured by great circle arc length. When I say convex hull of a subset $A\subseteq\mathbb{S}^n$, I do not mean convex hull in the Euclidean space containing $\mathbb{S}^n$; I mean the smallest subset $C_A$ of $\mathbb{S}^n$ that contains $A$ and such that, for any pair of points in $C_A$, any shortest path in $\mathbb{S}^n$ (i.e., piece of a great circle of length at most $\pi$) joining those two points is contained in $C_A$. To belabour the point, the unit ball is completely out of scope here; everything is a subset of $\mathbb{S}^n$. This is not Euclidean geometry; perhaps I picked the wrong tags?