I am trying to use the coarea formula to get estimates on the measure of an epsilon-neighbourhood of a set. Specificly, given a compact 'nice' set $A\subseteq \mathbb{R^d}$, possibly with more than one connected components which are not convex, I'm hoping to get an upper estimate on $$\lambda(A^\epsilon)-\lambda(A),$$
where $\lambda$ is the Lebesgue measure on $\mathbb{R}^d$ and $A^\epsilon= \cup\_{a\in A} B\_\epsilon(a)$. The notion of niceness is something that should be not just convex. I'm pretty sure it should be possible to write
$$ \lambda(A^\epsilon)-\lambda(A)= \int\_0^\epsilon \nu(A^{(t)})dt $$
by a Fubini like argument, where $A^{(t)}=\{x: d(x,A)=t \}$ and $\nu$ is some measure. I found [this thread](https://mathoverflow.net/questions/291561/relative-volume-increase-of-delta-fattening-of-a-compact-set) and related threads, which seem to rely on the Minkowski-Steiner formula. Is there perhaps some results of this nature using the co-formula area for non convex sets?



Coarea formula for measure of epsilon neighbourhood
Your formula is true (up to appropriate constants) if you take $\nu$ to be the Minkowski content, assuming that the volume of the tubular neighborhood is locally Lipschitz as a function of the tube radius. This is because the Minkowski content is defined as the derivative of the volume of the tubular neighborhood.


One condition ensuring that the tube volume is locally Lipschitz is that the distance function to $A$ has Clarke gradient bounded below in the $\epsilon$-neighborhood of $A$. It also ensures that Minkowski contents coincide with the Hausdorff measures of the boundaries.


Finally, although that's not the question, it is possible to bound the volume of the tubular neighborhood without any regularity conditions, using the $\epsilon$-covering numbers of $A$.


