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? 



Haar measure on the Grassmannian space
Yes, this is true. The answer by Ben Webster gives the idea, but for completion let me give you a list of easy facts that combined together answer the question.


1. The inequalities you seek are correct for any Riemannian manifold with bounds on the curvature (upper/lower bounds for corresponding sides of the inequalities), once you take the **Riemannian volume** and **Riemannian metric**. In particular, this is the case for a compact Riemannian manifolds.
2. Our Grassmanian could be seen as a Riemannian manifold, by embedding it in $\text{End}(\mathbb{R}^n)$ as you do, and restrict the standard inner product to the tangent spaces. The obtained Riemannian structure will be $G$-invariant (for $G=\text{O}(n)$ of course).
3. For our manifold, up to normalization, the Riemannian volume coincides with the Haar measure, as both are $G$-invariant, by "uniqueness".
4. For a Riemannain manifold $M$ and a compact submanifold $N$ the two metrics on $N$, the restriction to $N$ of the Riemannian metric of $M$ and the Riemannian metric of $N$ associated with the induced Riemannian structure, are biLipischits (this is an easy excercise).
5. All norms on a Euclidean space are equivalent.




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Of course, we apply 5 for whatever norm you had in mind in the formulation of the question and the standard inner product on $\text{End}(\mathbb{R}^n)$ and we apply 4 with $M=\text{End}(\mathbb{R}^n)$ and $N$ being the image of the Grassmannian.


