Antipodal-free closed cover from simplex Voronoi cells ($C_i=\{x\in S^n:x\cdot v_i\ge x\cdot v_j\text{ for all }j\}$)

Let $v_0,\dots ,v_{n+1}$ be the vertices of a regular simplex centered at zero in ${\mathbb{R}}^{n+1}$. The displayed spherical Voronoi cells are a closed cover by $n+2$ sets. If $x$ and $-x$ belonged to the same cell, all their scalar products with the vertices would coincide. Their sum is zero and the vertices span the ambient vector space, so this would force $x=0$, a contradiction. Thus each cell contains no antipodal pair. Together with the Lusternik-Schnirelmann-Borsuk theorem, this shows $n+2$ is the minimum number of antipodal-free members of a finite closed cover.