Compact set of the plane with boundary of finite 1-Hausdrof meausre as intersection of open set with piewise smooth boundary.

Question

I am considering a compact set $K$ of the plane with $\mathcal H^1(\partial K)=1$. Does there exists open sets with smooth boundary $O_n$, $n\in \mathbb N$ such that $K=\bigcap_n O_n$ and $\mathcal H^1(\partial O_n)\xrightarrow{n}1$? I guess this should be true (with hands i think that I manage to show $\limsup_n\mathcal H^1(\partial O_n)\leq 100$?). For the lower semicontinuity one may certainly use Golab's theorem (but i am not assuming there that $K$ is connected).

What about the same problem in higher dimensions $d>2$? Assuming maybe $\partial K$ is also $d-1$rectifiable?

Many thanks for the help!

Answer 1

I found some positive answer in Theorem 1.24 of Minimal surfaces and functions of bounded variations by Giusti.