I am given a smooth Riemannian submanifold of dimension $d$ embedded in $\mathbb{R}^D$ with condition number $1/\tau$ (a formal definition of condition number is on Page 3 of this paper http://people.cs.uchicago.edu/~niyogi/papersps/NiySmaWeiHom.pdf).
I would like to know how to calculate an upper bound on the $d$-dimensional volume of a ball (of radius $r \ll \tau$) around a point on the manifold intersected with the manifold. Lemma 5.3 in the paper referenced above gives a lower bound on this volume.
Proposition 6.3 gives a bound on the geodesic distance $d_M$ between any two points that are a distance $r$ apart in $\mathbb{R}^D$ as $$d_M(p,q) \leq \tau - \tau \sqrt{1 - \frac{2r}{\tau}} := d_{\max}$$ if $r < \frac{\tau}{2}$.
I think a valid upper bound is given by $$\mathrm{vol}(B(r,p) \cap M) \leq v_d d_{\max}^d$$ where $v_d$ is the volume of a unit ball in $\mathbb{R}^d$. How would I go about proving this (or an equivalent statement)?