Timeline for Monotonicity of doubling dimension
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 9, 2021 at 12:16 | vote | accept | SetValued_Michael | ||
| Aug 9, 2021 at 12:17 | |||||
| Aug 5, 2021 at 14:37 | comment | added | user142382 | As this example made me somewhat nervous (the Assouad dimension is not monotone???) let me say what allayed this: If one defines the Assouad dimension in a more careful way (as is done in Assouad's original paper, or in two equivalent ways in and after Def. 10.15 of Heinonen's Lectures on Analysis on Metric Spaces), then the dimension is indeed monotone. But it seems that then it does not necessarily exactly agree with log_2(doubling constant). | |
| Aug 3, 2021 at 21:46 | comment | added | Del | Sorry I deleted by mistake my previous comment trying to edit it. It was a particular case of the last one I wrote | |
| Aug 3, 2021 at 21:44 | comment | added | Del | @PiotrHajlasz I do think there may be some bad example, where the constant increases. The following example came to mind: $\mathbb{R}^d$ with the sup norm has doubling constant $2^d$ (at least if you consider closed balls...) but the subset given by all points whose coordinates belong to $\{-1,0,1\}$ should have constant $3^d$ (consider the ball centered at 0 with radius slightly above 1) | |
| Aug 3, 2021 at 21:41 | comment | added | Piotr Hajlasz | Good example. That explains everything. | |
| Aug 3, 2021 at 20:37 | comment | added | Piotr Hajlasz | Yes, but you get a quite bad estimate for the constant. Can you get the same constant for the subset or at least $C_X+\epsilon$? I haven't thought about it. | |
| Aug 3, 2021 at 20:16 | history | answered | Del | CC BY-SA 4.0 |