Convex hull of points in $\mathbb{S}^n$ not contained in a hemisphere

Question

Suppose I have $n+2$ points in $\mathbb{S}^n$ in the induced metric. In this question, the asker states that the convex hull of any four points in $\mathbb{S}^2$ which are not contained in a hemisphere must be all of $\mathbb{S}^2$ (they give a citation, but I don't have access to it). I have tried to prove it myself, but I haven't made much progress. How is this result proved? Also, I suspect that that a similar result holds in higher dimensions. Is this the case?

Answer 1

Partial answer: the result continues to hold in higher dimensions. Several authors (Danzer, Grünbaum and Klee, Helly's theorem and its relatives, citing e.g. Robinson, Spherical theorems of Helly type and congruence indices of spherical caps and Blumenthal, Metric methods in linear inequalities) assert without proof that convex sets (in the sense that every pair of non-antipodal points are connected by a minimising geodesic, the first reference also uses some other definitions) are intersections of hemispheres.