Timeline for Distance function and geometry of the set
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 17, 2020 at 0:54 | vote | accept | Lostsoul | ||
| Nov 9, 2020 at 9:20 | answer | added | mlk | timeline score: 0 | |
| Nov 3, 2020 at 21:13 | comment | added | mlk | By drawing some circles I think I am able to show that the tangent space of $\partial X$ in any point exists and is equal to the $x_n$-hyperplane. This should imply the form that Pietro is suggesting, but there might be some weirdness that I am missing, e.g. if F is something like the Cantor set. | |
| Nov 3, 2020 at 12:36 | comment | added | Lostsoul | Right, I agree with that claim. The proof or some hints about X being a union of hyperplanes would be very helpful. | |
| Nov 3, 2020 at 7:08 | comment | added | Pietro Majer | I'm saying $X$ is a product $F\times\mathbb{R}^{n-1}$ for a closed $F\subset\mathbb R$, without assuming $X$ is a $d$-dimensional regular set. I'll try to write a proof later. | |
| Nov 3, 2020 at 2:35 | comment | added | Lostsoul | The closest point is not always unique. There could be two, the $z_{+}$ and $z_{-}$. A zone you have mentioned won't be $d$-regular though unless I am missing something. Any suggestions why this condition necessarily implies that $X$ is a union of parallel hyperplanes? It seems intuitive but can't come up with a proper argument. | |
| Nov 3, 2020 at 0:17 | comment | added | Pietro Majer | And if the closest point is assumed to be always unique, $X$ is a zone $\{a\le x_n\le b\}$ for some $a\le b$ | |
| Nov 3, 2020 at 0:07 | comment | added | Pietro Majer | It seems to me that the only assumption on the form of the closest points already implies that $X$ is a union of parallel hyperplanes orthogonal to $e_n$. | |
| Nov 2, 2020 at 23:56 | comment | added | Pietro Majer | It is not clear to me if the closest point is assumed to be always unique. | |
| Nov 2, 2020 at 1:12 | review | First posts | |||
| Nov 2, 2020 at 4:04 | |||||
| Nov 2, 2020 at 1:05 | history | asked | Lostsoul | CC BY-SA 4.0 |