Timeline for $(1+\epsilon)$-bilipschitz parametrization of Lipschitz manifold
Current License: CC BY-SA 4.0
17 events
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| Sep 1, 2022 at 11:19 | vote | accept | No-one | ||
| Aug 30, 2022 at 14:57 | answer | added | No-one | timeline score: 1 | |
| Aug 30, 2022 at 11:46 | comment | added | Vladimir Zolotov | @Titti yeah. Staircase should work. Maybe you should post as answer so it's more visible. | |
| Aug 29, 2022 at 23:36 | comment | added | No-one | I think that the Volterra function or the "staircase" function of the fat Cantor set are counter-examples to my question. See math.stackexchange.com/questions/3380703/… | |
| Aug 29, 2022 at 23:12 | comment | added | No-one | @LeoMoos Do you know how to overcome the difficulty that even if $f'(0)=0$, a priori $f'$ could be large in any neighborhood of $0$ since we don't have continuity? Only thing that comes to my mind is the Lebesgue differentiation theorem but it doesn't seem to help much. | |
| Aug 29, 2022 at 23:07 | history | edited | No-one | CC BY-SA 4.0 |
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| Aug 29, 2022 at 22:05 | comment | added | Leo Moos | I think that with the most recent edit in mind - namely, that $M$ is assumed to be a Lipschitz graph - the desired result holds at the points where $f$ is differentiable. This would have full measure by Rademacher's theorem. | |
| Aug 29, 2022 at 19:51 | comment | added | No-one | @AntonPetrunin What do you mean by double twist? Do you know of any paper that does this construction in a more detailed way? Thanks. | |
| Aug 29, 2022 at 19:48 | answer | added | Vladimir Zolotov | timeline score: 2 | |
| Aug 29, 2022 at 19:25 | history | edited | No-one | CC BY-SA 4.0 |
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| Aug 29, 2022 at 19:11 | history | edited | No-one | CC BY-SA 4.0 |
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| Aug 29, 2022 at 18:36 | comment | added | Anton Petrunin | An counterexample exists for $m=1$ and $n=2$. Take a smooth curve and add to it double twists at sufficiently dense set of point, it can be done by keeping the curve smooth and bilipscitz with fixed constants + the increase of length can be arbitrary small. Then repeat the operation for smaller twists at even denser set of points + pass to the limit. | |
| Aug 29, 2022 at 15:12 | comment | added | Vladimir Zolotov | It seems to me that $(1+\epsilon)$-biLipshitz corresponds to the situation when you have $Lip(u) < \epsilon$ bound. (And I think there is a counterexample for this.) | |
| Aug 29, 2022 at 13:26 | history | edited | No-one | CC BY-SA 4.0 |
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| Aug 29, 2022 at 13:23 | comment | added | No-one | crossposted to mathstackexchange math.stackexchange.com/questions/4520925/… | |
| Aug 29, 2022 at 12:45 | history | edited | No-one | CC BY-SA 4.0 |
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| Aug 29, 2022 at 12:28 | history | asked | No-one | CC BY-SA 4.0 |