Timeline for Is the Lipschitz constant of $f$ equal to $\|\nabla f\|_{L^\infty}$?
Current License: CC BY-SA 4.0
9 events
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Jul 25, 2023 at 23:58 | comment | added | Nik Weaver | *argue that $|f(p)-f(q)| \leq$ the derivative of $f$ restricted to that line times $d(p,q)$ | |
Jul 25, 2023 at 19:16 | comment | added | Nik Weaver | (I doubt this is my argument, but if not I don't remember where it's from.) | |
Jul 25, 2023 at 19:15 | comment | added | Nik Weaver | That doesn't quite work because the derivative might not exist a.e. on every line, but you can thicken that line and integrate over a ball lying in the hyperplane perpendicular to the line joining $p$ and $q$. | |
Jul 25, 2023 at 19:14 | comment | added | Nik Weaver | @MikhailKatz Oh, it's pretty elementary. The $\leq$ part is easy, in the opposite direction pick any two points, $p$ and $q$, draw a line joining them, and attempt to argue that $d(p,q) \leq$ the derivative of $f$ restricted to that line, in absolute value. | |
Jul 25, 2023 at 16:43 | comment | added | Mikhail Katz | Hi Nik! Do you prove that via Sobolev spaces, or by a more elementary argument? | |
Jul 25, 2023 at 16:38 | comment | added | Nik Weaver | I should have specified "second edition". Edited. | |
Jul 25, 2023 at 16:38 | history | edited | Nik Weaver | CC BY-SA 4.0 |
added 17 characters in body
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Jul 25, 2023 at 16:31 | vote | accept | Akira | ||
Jul 25, 2023 at 16:27 | history | answered | Nik Weaver | CC BY-SA 4.0 |