What is the covering number $N(\epsilon, B\_2, ||\cdot||\_2)$ of a ball $B\_2$ in $\mathbb{R}^d$ of radius $r$ under the $l\_2$ norm?



Covering number of $l_2$ Ball in $\mathbb{R}^d$
The $\epsilon$-covering number of the euclidean unit ball in $\mathbb R^d$ scales like $(1/\epsilon)^d$. More formally,



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> **Lemma.** If $\epsilon < 1$, then $(1/\epsilon)^d \le N(\epsilon, B\_2) \le (3/\epsilon)^d$. Else $N(\epsilon,B\_2) = 1$.
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*Proof.* See **Theorem 4.2** and **Example 14.1** of this manuscript <http://www.stat.yale.edu/~yw562/teaching/598/lec14.pdf>. $\quad\quad\Box$


Now, use this to get (an estimate of) the covering number of $rB\_2$, for any $r \ge 0$.


