Timeline for Why a function induced by the infimum of the arclength of curves is Lipschitz?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6 at 14:58 | comment | added | Javier | Thank you, @TaQ! | |
| Jul 6 at 0:33 | comment | added | TaQ | If $y_1$ and $y_2$ can be joined inside $\Omega$ by line segment not meeting $S_i\,$, then for every (rectifiable) curve going inside $\Omega$ from $x_0$ to $y_1$ there is the curve $x_0\to y_1\to y_2$ with the same length of the part that is inside $S_i\,$, and conversely. So the sets whose infimums are $l_i(y_1)$ and $l_i(y_2)$ are the same, and so $l_i(y_2)=l_i(y_2)$ holds. Every point in $\Omega\setminus{\rm Cl\,}S_i$ has an open neighbourhood $N$ such that the above holds for all $y_1,y_2\in N\,$. Does this make sense to you? | |
| Jul 5 at 7:36 | comment | added | Javier | Thank you, @TaQ! I understand $(*)$, since differentiability is a local property; hence, locally Lipschitz is enough for us to desire $(*)$. But I am not sure about your comment on 3. Could you please write the precise process of it? Thank you! | |
| Jul 3 at 17:32 | comment | added | TaQ | Your observation "I am sure that $l_i$ is locally Lipschitz ..." is correct. For general $\Omega$ an assertion about being globally Lipschitz is false. Neither is it needed here. As Nate River already noted, assertion $(*)$ is a trivial consequence of being locally $1\,$−Lipschitz. Point 3. is false. There should be the *closure* of $S_i\,$, noting that $l_i$ is locally constant outside ${\rm Cl\,}S_i$ almost directly by its definition. | |
| Jul 1 at 8:27 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Added reviews + minor formatting
|
| Jul 1 at 6:15 | history | edited | Javier | CC BY-SA 4.0 |
added 47 characters in body
|
| Jun 30 at 15:08 | comment | added | Javier | Sorry, I extracted one sentence from the paper, and it seemed confusing. Therefore, I deleted this sentence and added a full description at the end. @NateRiver | |
| Jun 30 at 15:04 | history | edited | Javier | CC BY-SA 4.0 |
added 1055 characters in body
|
| Jun 30 at 14:03 | comment | added | Nate River | Concerning your question 2, a point at which $|\nabla l_i| > 1$ would not be a point of local Lipschitz constant $1$ of $l_i$. | |
| Jun 30 at 13:57 | comment | added | Nate River | Sorry, what is meant by $x_0$ lies in different connected components of $\Omega \setminus \bar S_i$? | |
| Jun 30 at 12:37 | history | edited | Javier | CC BY-SA 4.0 |
deleted 4 characters in body
|
| Jun 30 at 12:21 | history | asked | Javier | CC BY-SA 4.0 |