TITLE: Covering a sphere with spherical caps
QUESTION [5 upvotes]: Consider the $n-1$ dimensional unit sphere embedded in $\mathbb{R}^n$. For example, when $n=3$, the sphere is characterized by $x_1^2+x_2^2+x_3^2=1$. Define a special point as a point whose coordinates are all zero except one coordinate that is either $1$ or $-1$. For example, if $n=3$, there are six special points: $(1,0,0)$, $(-1,0,0)$, $(0,1,0)$, $(0,-1,0)$, $(0,0,1)$, $(0,0,-1)$. In general, there are $2n$ special points in $\mathbb{R}^n$.
Consider a spherical cap centered at each special point. So there are $2n$ caps as well. Suppose all caps have the same height $h$. I am interested in finding the smallest $h$ such that allows these caps to entirely cover the sphere.
Thank you!
Golabi

REPLY [7 votes]: Consider the cube with vertices $(\pm 1, \ldots ,\pm 1)$ in $\mathbb{R}^n$. It is inscribed in the sphere of radius $\sqrt{n}$ centered at the origin. Note that if we consider hyperplanes parallel to the faces of the cube then the corresponding caps on the sphere cover the whole sphere (if you slice the sphere along those planes you are left with the cube). However, if we took the caps higher than the faces then the vertices of the cube are left out. The ratio we want 
$$\frac{\text{ height of cap}}{\text{ radius of sphere} }= \frac{\sqrt{n}-1}{\sqrt{n}}=1- \frac{1}{\sqrt{n}}$$
like the previous answer