1) Can the sum of two non convex set be a convex set ?
2) Can the sum of convex set and non convex set be a convex set ?
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1For #2, what is the sum of $[0,1]$ and $\Bbb Z$? – Cameron Buie Feb 16 '13 at 17:31
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OK, let's make the comment an answer:
Take $A=\mathbb R\setminus\{0,1\}$. Note that $A$ is not convex. Let $B=[0,2]$, which is convex. We have $A+B=\mathbb R$ which, of course, is also convex: $$ A+B\supseteq A+\{0,2\}\supseteq(\mathbb R\setminus\{0,1\})\cup\{-2+2,-1+2\}=\mathbb R. $$
If we want $B$ to be non-convex as well, use the same $A$ and now take $B=\{0,2\}$.
Andrés E. Caicedo
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In your first example I don't think $2$ or $3$ are in $A+B$. I think you just need to take $B=[0,2]$ or something. – JSchlather Feb 16 '13 at 18:49
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