What is the geometric interpretation of this quantity? Let $(M,g)$ be  a compact $n$-dimensional Riemannian manifold. Using the metric to identify the tangent and cotangent bundles defines a natural symplectic 
structure on the tangent bundle, $(TM, \omega)$, and a natural volume form $\Omega=\omega^{n}$ on $TM$.
For every $r>0$, let $D_{r}(M)$ be the open disc bundle on $M$, with radius $r$. Namely $D_{r}(M)$ is the subset of $TM$ formed by all tangent vectors with length smaller than $r$.
Define:
${C(r)=(\text{Gromov width of $D_{r}(M)$}})^{2n}$
$V(r)$=The Volum  of $D_{r}(M)$ with respect to $\Omega$
Question :

1) Does  $\lim_{r\to \infty} C(r)/V(r)$ exist?  And what is its geometric interpretation?
2) Does   $\lim_{r\to 0} C(r)/V(r)$ exist?    And what is its geometric interpretation?

Recall that the Gromov width of  a symplectic manifold $N$ of dimension 2n defined as follows:
$\sup\; \{\rho \mid \text{there is  a symplectic embedding from $B_{\rho}(0)\subset \mathbb{R}^{2n}$ to $N$}\}$.
By $B_{\rho}(0)$  I mean the disc around the origin with radius $\rho$.
 A: Both limits (1) and (2) are equal to $C(1)/V(1)$ because of the homogeneity of the volume and the symplectic capacity. Namely, the symplectic form is homogeneous of degree $1$ with respect to dilations on the tangent bundle and so it makes no difference what the radius of your disc bundle is.
What is then geometric meaning of $C(1)/V(1)$ ? That's an interesting and difficult question which falls under the topic of capacity-volume inequalities (something that is just taking off). I'd suggest looking at flat tori first. The reason is that if you consider the universal cover, the lattice, and the ellipsoid that defines your unit disc, then if you have a lattice basis inside the ellipsoid, you also have a symplectic unit ball inside the unit tangent bundle of your torus : just look at the moment map of the torus action on the unit ball: it's image is (or can be made to be) the simplex formed by the origin and your lattice basis. You will probably be able to give a number-geometric interpretation to $C(1)/V(1)$ in this case.
By the way, the original capacity-volume problem is
The Viterbo Conjecture. The symplectic capacity (any capacity!) of a convex body $K \subset \mathbb{R}^{2n}$ whose volume is the same as that of the $2n$-dimensional Euclidean unit ball is less than or equal to $\pi$.
Note that this is obviously true for the Gromov width, but it is open for every other capacity. There is a chance this conjecture is true for the Hofer-Zehnder capacity, which can be more easily described as the least of the actions of all closed characteristics on the boundary of $K$. There are some partial results on this conjecture in the first version of this paper, but beware, by a recent result of Artstein-Avidan, Ostrover, and Karasev, a positive answer to Viterbo's conjecture would also prove the Mahler conjecture (corollary: this can't be easy!).
