A volume comparison theorem for Finsler manifolds

Let $(M^{n},F)$ be a symmetric Finsler manifold, endowed with the Busemann volume form, and let $D$ be its unit disk bundle endowed with the canonical symplectic volume form. It is shown that $Vol(D)\leq C(n)Vol(M^{n})$, where $C(n)$ is the volume of the unit disk in ${\mathbb{R}}^{n}$. Moreover, equality holds if and only if $(M^{n},F)$ is Riemannian.