Home > Geometry, shape optimization > Sets of finite perimeter which do not contain any open ball

Sets of finite perimeter which do not contain any open ball


Some time ago, I wondered myself if I could find a very small ball included in a set of finite perimeter (see definition for set of finite perimeter here and here). It turns out, that the answer is no, there mustn’t be any open ball included in a set of finite perimeter. My teacher gave me the following nice example.

Example: Take { D=B(0,1)}, the unit ball in { \Bbb{R}^2} and denote

\displaystyle S=D\cap \mathbb{Q}^2 = (x_n)_{n \geq 0}

. Then, we can find a sequence of positive real numbers {(r_n)} such that

\displaystyle \sum_{n=1}^\infty 2\pi r_n < \infty

\displaystyle \sum_{n=1}^\infty \pi r_n^2 < \pi

Define \displaystyle B=\bigcup_{n=1}^\infty B(x_n,r_n).
Then {C=D\setminus B} has the desired property. Indeed, {|C|=|D|-|B|>0}, and {Per_D(C)=Per_D(B)<\infty}.

If {C} would contain an open ball then that ball would intersect {S \subset B}, which is not possible.

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