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?


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