(A version of this question was posted on math stack exchange)
Let $M$ be a $C^1$ submanifold of dimension $n$ of $\mathbb{R}^N$, and denote $\mu$ the standard surface measure on $M$.
Consider a bounded subset $S$ of $\overline M$ with Hausdorff dimension $k\leq n$ (if it helps suppose $S$ is a submanifold of dimension $k$), and denote for all $\epsilon>0$,
$T(\epsilon):=\{x\in \mathbb{R}^N : d(x,S)<\epsilon\}$ the tube around $S$ at distance $\epsilon$ (here $d$ is a distance associated to a norm on $\mathbb{R}^N$).
Question : do we have $\mu(T(\epsilon)\cap M)=\mathcal{O}_{\epsilon\to 0}(\epsilon^{n-k})$ ?
Motivation : On simple examples, this seems to hold. For example, if $S$ is just a point in the plane, then we see indeed that the $\epsilon$-tube is a disk of radius $\epsilon$, and so its area will decay as $\epsilon^2$. For another example, in $M=\mathbb{R^3}$ if we take $S$ as a circle, the $\epsilon$-tube will be a filled torus with volume decaying as $\epsilon^2$. I am wondering how this behavior can be generalized.
