Number of balls covering a $n$-dimensional sphere

Question

What upper bounds for number of covering a unit sphere balls with a same radius $r$ and centers on this sphere are known?

Answer 1

I'll give a reference. Have a look at Lemma 3 in Section 1.4.3 "Maximising Over the Euclidean Ball". (The numbering is poorly done; Lemma 3 is at the end of page 10.) That gives (almost) exactly what you want. (It's a lecture course I'm doing at the moment.)

This gives the proof that it's the number of balls of size $r$ required to cover the whole ball is less than or equal to $(3/r)^n$. Note, however, the stark difference in high-dimensional geometry: for large $n$, the volume concentrated around the boundary is where almost all the volume of the ball is. Indeed, suppose you have constants chosen such that the volume is $r^n$. Then $(\frac{r}{0.99r})^n \to \infty$ and $r^n - (0.99r)^n \to \infty$ as $n \to \infty$.