TITLE: Intersections of connected sets with piecewise smooth boundaries
QUESTION [2 upvotes]: Suppose you have two connected sets $S_1$ and $S_2$ in $\mathbb{R}^n$ with piecewise smooth boundaries, and whose intersection $S=S_1 \cap S_2$ has positive Lebesgue measure. Will the sets $S$, $S_1 \backslash S_2$ and $S_2 \backslash S_1$ all have smooth boundaries, and if yes: how can I prove this?

REPLY [1 votes]: My understanding of piecewise smooth boundaries is that they consist of finitely many smooth surfaces. In that case, the answer is no. In two dimensions, let $S_1=\{(x,y):y> \sin x\}$ and   $S_2=\{(x,y): y>0\}$. Then the boundary $S_1\cap S_2$ has infinitely many corners. 
Alternatively, use $S_1=\{(x,y):y> x\sin (1/x)\}$ and the same $S^2$; then the corners will have a point of accumulation. And if you  replace $x\sin (1/x)$ with $\exp(-1/x^2)  \sin (1/x)$, then the boundary of $S_1$ will be smooth, not just piecewise smooth.
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