Suppose that $M\subseteq\mathbb R^D$ is a compact smooth Riemannian submanifold of dimension $d$, having normal injectivity radius $\tau$. Let $x\_0\in M$ be a point, and $\delta\in (0,\tau)$ sufficiently small. I would like to know if $M\cap B\_{\mathbb R^D}(x\_0,\delta)$ has $d$-dimensional Hausdorff measure at least $(1/2)\delta^d\omega\_d$ where $\omega\_d$ is the volume of unit ball in $\mathbb R^d$.


My approach: since $\delta\in (0,\tau)$, there is a diffeomorphism $\exp\_{x\_0}:B\_{T\_{x\_0}M}(0,\delta)\cong B\_M(x\_0,\delta)$. Now, by Taylor expansion of the volume form (theorem 9.9/9.10 of Gray's *Tubes*) we know, for example, that volume of $B\_M(x\_0,\delta)$ is more than $0.9$ times the volume of $B\_{\mathbb R^d}(0,\delta)$; hence, if I know that $B\_M(x\_0,\delta)\subseteq M\cap B\_{\mathbb R^D}(x\_0,\delta)$, for example, then I will be done.



largest geodesic ball inside a small portion of Euclidean submanifold
For all $x,y \in M$, $d(x,y) \geq \lvert x - y \rvert$, where $d$ is the distance function on the manifold. Therefore for every point $x\_0 \in M$ and every radius $\rho > 0$,
\begin{equation}
B\_M(x\_0,\rho) = \{ x \in M \mid d(x,x\_0) < \rho \}
\subset \{ x \in M \mid \lvert x - x\_0 \rvert < \rho \}.
\end{equation}
This holds in particular for small radii, for example when $\rho = \delta$ in your description.


