Timeline for Bound measure of difference of advected sets by norm of difference of vector fields
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2 at 21:37 | comment | added | alesia | also, if you don't want to assume regularity, you can bound the volume of tubular neighborhoods using covering numbers of the boundary (consider union of balls) | |
| Jul 2 at 21:23 | history | edited | alesia | CC BY-SA 4.0 |
deleted 7 characters in body
|
| Jul 2 at 21:21 | comment | added | alesia | yes the idea is essentially as @unwissen describes. The tube formula provides an exact formula refining the above area based first order expansion, which is useful if you want an upper bound and not a first order behavior | |
| Jul 2 at 17:43 | comment | added | unwissen | Just to have my understanding verified, not because I think I can explain @alesia's nice idea better: the measure of all points which have distance less than $\epsilon$ to $\Omega_u$ (but are not included in $\Omega_u$) is $\lesssim \epsilon \vert \partial \Omega_u \vert \lesssim \epsilon \vert \partial \Omega \vert $. Therefore the measure of all points which are in $\Omega_v$ but not in $\Omega_u$ is $\lesssim d_H(\Omega_u, \Omega_v) \vert \partial \Omega \vert$. By symmetry, $\vert \Omega_u \Delta \Omega_v \vert \lesssim d_H(\Omega_u, \Omega_v) \vert \partial \Omega \vert$. | |
| Jul 2 at 16:57 | comment | added | tommy1996q | Could you elaborate a little? In particular bounding the volume using the Hausdorff distance and the argument using the tube formula and the reach (never heard of neither of them to be honest) | |
| Jul 2 at 16:17 | history | answered | alesia | CC BY-SA 4.0 |