The boundary of this set is smooth?

Question

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 !

Answer 1

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 $$ f_i(x,y) = (|x|^i + |y|^i)^{\frac 1 i}. $$