# Jaggedness of the boundary of the $\epsilon$ neighborhood of a set

Let $$X$$ be a simply connected set in $$\mathbb{C}$$. Of course $$X$$ can be quite jagged and nasty. Let $$\epsilon>0$$ be given, and define $$C(X;\epsilon)=\{z\in\mathbb{C}:\min(|z-w|:w\in X)=\epsilon\}$$.

Essentially one my find $$C(X;\epsilon)$$ by taking a radius-$$\epsilon$$ ball, and rolling it around the outside of $$X$$, and tracing the motion of the center of the ball (I know this ignores possible extra components of $$C(X;\epsilon)$$, which is ok with me for the purpose of this question).

My impression is that $$C(X;\epsilon)$$ smooths out the boundary of $$X$$ quite a bit.

1) Can this be quantified? For example, it seems to me that $$C(X;\epsilon)$$ must be at least rectifiable, parameterized by a closed path $$\gamma:[0,1]\to\mathbb{C}$$. Are there better quantifications?

2) If this is quantified in a meaningful way, will $$C(C(X;\epsilon);\epsilon)$$ be any smoother than $$C(X;\epsilon)$$? My guess is no.

3) If one first forms $$C(X;\epsilon)$$ and then in some appropriate way shrinks this set by a distance of $$\epsilon$$ back towards the boundary of $$X$$, it seems like this will be a nice smoothing approximation of the boundary of $$X$$, getting ever better as $$\iota\to0$$. Is this method of approximating known to anyone?