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Let $\Omega_1 \supset \Omega_2 \supset....$ a decreasing sequence of bounded, convex and smooth sets.

My intuition says that the set $int(\overline{\bigcap_i \Omega_i})$ (where int denotes the interior of a set) has smooth boundary. I dont know how to prove or disprove this ...

Someone can give me a help ?

thanks in advance !

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  • $\begingroup$ When you say smooth, how smooth should it be? $C^\infty$, $C^n$, Lipschitz, cone condition, segment condition? Please be more specific. $\endgroup$ – Yiorgos S. Smyrlis Feb 17 '14 at 13:25
  • $\begingroup$ @Yiorgos S. Smyrlis , smooth = $C^{\infty}$. Sorry to not say this. $\endgroup$ – math student Feb 19 '14 at 0:55
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The intersection of your sets might be any convex set. In particular it might be a square, which is not smooth.

To get a square consider $$ \Omega_i = \{(x,y)\in \mathbb R^2 \colon f_i(x,y) < 0\} $$ where $$ f_i(x,y) = (|x|^i + |y|^i)^{\frac 1 i}. $$

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