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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
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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