Let $M$ be a compact path metric space in $\mathbb{R}^d$, and for $\sigma>0$,
$$
M\_\sigma:=\{y\in\mathbb{R}^d:\min\_{x\in M}\|x-y\|\leq\sigma\}
$$
the $\sigma$-tube around $X$ in $\mathbb{R}^d$. I consider both $M$ and $M\_\sigma$ metric spaces with respect to the shortest path metric (geodesic, not necessarily Euclidean distances) induced by $\mathbb{R}^d$, with possibly distinct intrinsic dimensions. We are furthermore given constants $s,\epsilon>0$, such that for $x,y\in M$, $\|x-y\|<s\implies d\_M(x,y)<\epsilon$. Is there any bound we can provide on the Gromov Hausdorff distance $d\_{GH}(M, M\_\sigma)$ in terms of $s,\epsilon$, and the diameter of $M$, when $\sigma$ is sufficiently small?


The tubular neigborhood can significantly alter the metric, e.g., the tubular neighborhood of a nearly closed circle can suddenly include the circle itself. However, I suspect that such information would be encoded by $s$ and $\epsilon$, and that for $\sigma$ sufficiently small (according to these parameters), the path from $x$ to $y$ in $M\_\sigma$ travels `near' the path from the (not necessarily unique) projections of $x$ and $y$ on $M$, and the length of these paths will then be similar.


I could believe that similar problems have been investigated before, but I don't find any helpful references. It would be great if someone could point out some possible directions on this problem.



Gromov Hausdorff distance to tubular neighborhood
I think I have figured this out. More specifically, it should hold that
$$
d\_{GH}(M, M\_\sigma) \leq \max\left\{2\sigma, \left(\frac{\epsilon}{s-2\sigma}-1\right)(\mathrm{diam}(M)+2\sigma)+\epsilon\right\},
$$
whenever $\sigma < s/2$.


Sketch of the proof:


Define the correspondence $C$ as
$$
(x,y)\in C\leftrightarrow y\in \overline{B}\_{\mathbb{R}^d}(x,\sigma)
$$
Clearly it holds that $(x,y),(x',y')\in C$ implies that $d\_{M\_\sigma}(y,y')\leq d\_M(x,x')+2\sigma.$
For the more difficult direction, take $0<\delta\leq s-2\sigma$ and split up the path from $y$ to $y'$ in $M\_\sigma$ into $k$ parts of length at most length $s - 2\sigma - \delta$. This can be done with $k\leq \frac{d\_{M\_\sigma}(y,y')}{s - 2\sigma - \delta}+1$ segments. Each of these segments corresponds to a segment to a segment in $M$ with length at most $\epsilon$. We find that $d\_M(x, x')\leq \left(\frac{d\_{M\_\sigma}(y,y')}{s - 2\sigma - \delta}+1\right)\epsilon$. Now subtract $d\_{M\_\sigma}(y,y')$ from both sides, and bound $d\_{M\_\sigma}(y,y')$ in the right hand side by $\mathrm{diam}(M\_\sigma)\leq\mathrm{diam}(M)+2\sigma$. Finally, let $\delta\rightarrow 0$.


