Regularity of the set of point with a small distance from the boundary Let $ \Omega $ be a bounded, open, connected subset of $\mathbb{R}^N$.
Let $\varepsilon > 0$ and define $\Omega_{-\varepsilon} = \{x \in \Omega: dist(x,\partial \Omega) > \varepsilon \}$.
What can we say about $\partial \Omega_{-\varepsilon}$? Does it have a greater regularity compared to $\partial \Omega$?
Suppose, for example, that $\Omega$ is a triangle in the plane. Then is $\Omega_{-\varepsilon}$ a smaller and smoother(without kinks) version of that triangle? If yes, is there a good reference on it?
Thank you very much for your help.
 A: You have not defined "greater regularity", or "regularity", but at any rate for a triangle $\Omega$ we have that $\Omega_{-\varepsilon}$ is a similar triangle (unless empty, f0r big enough $\varepsilon$). It does have exactly the same kinks as $\Omega$. 
Now start with an open triangle $T$ (with kinks), take any $\varepsilon>0$ and let 
$\Omega=\cup_{x\in T}B_\varepsilon(x)$. Then $\Omega$ has no kinks, but $\Omega_{-\varepsilon}=T$ does. 
So, in general $\Omega_{-\varepsilon}$ is clearly smaller, but not necessarily smoother. 
For a reference, you may study Minkowski addition (and subtraction), usualy treated in convexity books. $A+B=\{a+b:a\in A, b\in B\}$. If $B_\varepsilon$ is the ball centered at the origin or radius $\varepsilon$ in $\Bbb R^n$, then $\Omega_{-\varepsilon}=\Bbb R^n\setminus \bigl((\Bbb R^n\setminus\Omega)+
\overline{B_\varepsilon}\bigr)$. Note that $\Omega+B_\varepsilon$ is in general smoother that $A$. Also, $\Omega+B_\varepsilon=\Omega-B_\varepsilon=\{a-b:a\in\Omega, b\in B_\varepsilon\}$ (since $\varepsilon$ is symmetric about the origin, but $\Omega_{-\varepsilon}\neq\Omega-B_\varepsilon$. 
