Timeline for Is (the generalised) Sard's theorem optimal?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Sep 18, 2022 at 15:51 | comment | added | Sam Forster | Thank you for the link. That is a similar question, yet not quite what I'm looking for. | |
| Sep 18, 2022 at 15:43 | comment | added | David E Speyer | Okay. I wasn't sure whether you would be interested in examples of things like $C^1$ maps $\mathbb{R}^2 \to \mathbb{R}^1$ where $f(A_0)$ has dimension $1$. We discussed those here mathoverflow.net/questions/258141 . | |
| Sep 18, 2022 at 15:30 | comment | added | Sam Forster | @DavidESpeyer It's correct that Sard's theorem doesn't impose this. However since differentiable functions cannot increase Hausdorff dimension under images, the dimension of $f[\mathbb{R}^d]$ is at most $d$. But then since my question is asking whether $f[A_0]$ can have maximal dimension (which is $d$), it seems most restrictive to ask the question when the dimensions are the same. I want to know if the answer is positive in this restrictive case. | |
| Sep 18, 2022 at 15:20 | comment | added | David E Speyer | I'm pretty sure Sard's theorem didn't require the source and target to have the same dimension. (The math.SE link you quote doesn't.) Do you want to impose this? | |
| Sep 18, 2022 at 13:29 | history | asked | Sam Forster | CC BY-SA 4.0 |