I know there is a simple method that generates a sequence of closed shapes whose limit is the unit circle but the limit of the perimeter is not $2\pi$ , but in all of the cases that I know , the perimeter limit is finite.
How we can generate a sequence of shapes whose limit tends to unit circle but the limit of the perimeter is infinity?
Best Answer
In the annulus with inner radius $1-1/n$ and outer radius $1+1/n$ draw a curve that oscillates back and forth between the inner and outer edge often enough to have length $n$. Then the limit as $n \to \infty$ does what you ask for.