The grassmannian space $G(n,m)$ may be identified with the quotient space $O(n)/(O(m)\times O(n-m)$. As such, it is endowed with a natural invariant probability measure which I call "Haar measure on $G(n,m)$". Let $\gamma_{n,m}$ be this measure. Let us endow $G(n,m)$ with the usual metric $d(V,W)=\|P_V-P_W\|$, where $P_V$ is the orthogonal projection onto $V$.

**Question.** Is it true that there is some constant $C$ such that $\frac{1}{C} r^{m(n-m)} \leq \gamma_{n,m}(B(V,r)) \leq C\times r^{m(n-m)}$ for any $V \in G(n,m)$ and any $r>0$ small enough?