If A $\subset S^n$ has spherical diameter < $\pi$, then $S^n - A$ contains a closed semisphere.

Question

Let $S^n$ be the $n$-dimensional unit sphere in $\mathbb R^{n+1}$.

Let $X \subset S^n$ such that for any $x, y \in X$ the angle between $x$ and $y$ is smaller than $180°$.

Then $S^n - X$ does contain a semisphere, i.e., something isometric to $S^n \cap \{ x_{n+1} \leq 0 \}$. Can you give a proof with elementary tools and without directly using the Hahn–Banach separation theorem?

Answer 1

This is wrong: Let $X$ consist of the four vertices of a regular tetrahedron inscribed in $S^2$. Then the distance between any two points of $X$ is $<\pi$; but there is no hemisphere contained in $S^2\setminus X$.