= Antipodal-free closed cover from simplex Voronoi cells
{title2=$C_i=\{x\in S^n:x\cdot v_i\geq x\cdot v_j\text{ for all }j\}$}
Let $v_0,\ldots,v_{n+1}$ 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>.
Back to article page
