I am interested in bibliographical references about a special class of sets, those who have positive reach and which complementary has also positive reach.
I recall that the reach $R\geq 0$ of a set $A$ in $R^d$ is the largest number $r$ such that for every point $x$ at distance $r$ from $A$, there is a unique closest point of $A$ from $x$ (remark that this "projection" is on the boundary of $A$ if $x$ is not in $A$). They have primarily been defined and used in geometric measure theory.
I believe that those sets coincide with sets that are called Morphologically Open and closed (MOC) sets in mathematical morphology (image processing), which are the sets $A$ such that for some $r>0$, $(A\oplus B(0,r))\ominus B(0,r)=(A\ominus B(0,r))\oplus B(0,r)=A$ where the $\oplus$ and $\ominus$ operations are the classical erosion and dilation of sets:$$A\oplus B(0,r)=\{x:d(x,A)\leq r\}$$ $$A\ominus B(0,r)=\{x:B(x,r)\subset A\}.$$
I wonder if for instance each such MOC set can always be represented as the level set of a $\mathcal{C}^1$ function $f$: $A=f^{-1}(\{u\})$ for some $u\in R$.
I also saw in the works of Serra a proof of why the homeotopy of those sets sets are preserved under hexagonal lattice approximation. I am actually not fully satisfied with the proof.
I look for a detailed account on those sets, if any, and I would be more interested by works from people from geometric measure theory, or integral geometry, as the proofs from image processing lack sometimes a bit of rigour.
