I am trying to bound the covering number of the set $M$ of rank 1 matrices (in $\mathbb R^{n\times d}$) with frobenius norm at most 1. I can do this with the following method: given a $\delta$-cover $c_i$ of the set $M$ and a sequence $b_i$ of smaller $\delta/4$ balls centered at the $c_i$ we have the following relationship between their volumes:
$$\bigcup_i b_i \subset \bigcup_i c_i \implies \text{Vol}\bigcup_i b_i \leq \text{Vol}\bigcup_i c_i = N V_{\delta/4} \leq V_{1 + \delta/4}$$
Where $N$ is the covering number of $M$ and $V_r$ is the volume of the set of rank 1 matrices with norm at most $r$. So if I can find an expression for $V_r$ then I can bound the covering number.
It's not clear to me what the notion of volume means in this space. Some searching leads me to the idea of Haar measure, which I am unfamiliar with, but can understand as a measure over linear transformations.
my question
How do I compute $V_r$?
EDIT: bounding the covering number can be reduced to bounding the covering number of an $n+d$ euclidean ball, but I am still interested in the question I asked above.