I am trying to use the coarea formula to get estimates on the measure of an epsilon-neighbourhood of a set. Specificly, given a precompact 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 $A^\epsilon= \cup_{a\in A} B_\epsilon(a)$. 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 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?

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 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?