Timeline for Isoperimetric inequality for exterior domains on $\mathbb{H}^{n}$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2, 2021 at 18:20 | vote | accept | Shaq155 | ||
| Jul 2, 2021 at 14:08 | comment | added | mlk | @Shaq155 As the second part of the answer says, if you allow $\Omega$ to touch the boundary, the inquality in general no longer holds. | |
| Jul 2, 2021 at 12:58 | comment | added | Shaq155 | Sry for that confusion... | |
| Jul 2, 2021 at 12:58 | comment | added | Shaq155 | Thank you for your answer, but I actually need the Isoperimetric inequality for bounded domains... | |
| Jul 2, 2021 at 11:32 | comment | added | mlk | Assume that is not the case, then there are points $x_n \in \Omega$, $y_n \in K$ such that $d(x_n,y_n) \to 0$. As $K$ is compact, for a subsequence $y_n \to y \in K$ and as $\Omega$ is precompact in $\mathbb{H}^n \setminus K$, similarly $x_n \to x \in \overline{\Omega} \subset \mathbb{H}^n \setminus K$ for a subsequence of that subsequence. But then $d(x,y) = 0$, so $x=y$, which is a contradiction as $y\in K$ and $x \notin K$. | |
| Jul 2, 2021 at 11:28 | comment | added | Shaq155 | Could you explain why it has some distance, even though I guess this is trivial? | |
| Jul 2, 2021 at 11:24 | comment | added | mlk | @Shaq155 That part was the one thing that was clear. The detail that was initially missing was if $\Omega$ can come close to $K$. But since $\mathbb{H}^n \setminus K \subset \mathbb{H}^n$, if $\Omega$ is precompact in the former, it has some distance to $K$ and thus $\partial \Omega$ does not depend on which topology you use and the inequality then follows trivially. | |
| Jul 2, 2021 at 11:14 | comment | added | Shaq155 | Maybe my question was misleading, I consider domains $\Omega \subset \mathbb{H}^{n}\setminus K$. | |
| Jul 2, 2021 at 11:09 | comment | added | mlk | @Shaq155 In that case the measures of $\Omega$ and $\partial \Omega$ do not change if you consider them as subsets of $\mathbb{H}^n$ instead, so the inequality trivially holds. | |
| Jul 2, 2021 at 10:58 | comment | added | Shaq155 | I changed something. I consider $\mathbb{H}^{n}\setminus K$ as a manifold with boundary, that means $\partial \Omega\cap K=\emptyset$. | |
| Jul 2, 2021 at 10:22 | history | answered | mlk | CC BY-SA 4.0 |