Timeline for Gaussian isoperimetry for $\ell_p$ norms
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 21, 2019 at 12:12 | history | edited | dohmatob | CC BY-SA 4.0 |
added 4 characters in body
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| Mar 21, 2019 at 5:59 | comment | added | dohmatob | OK, this is more explicit. Thanks! Are there any references on a systematic treatment of the problem ($\ell_p$ Gaussian isoperimetry) ? Thanks! | |
| Mar 20, 2019 at 20:10 | comment | added | Alf | Yes, but not all halfspaces. Because the $\ell_p$ distances are not rotation-invariant for $p \neq 2$ not all halfspaces have the same blowup. | |
| Mar 20, 2019 at 18:49 | comment | added | dohmatob | @Jon Are we saying the extremal shapes $H$ for $\ell_p$ distances are also half-spaces ? | |
| Mar 20, 2019 at 18:47 | comment | added | dohmatob | @Dirk, Jon. Thanks for the input. Yes "equivalence of norms" can help get rough bounds (upper and lower) on the boundary measure of an $\ell_p$ blowup (and the dimension pops up in a perhaps sub-optimal manner). I am aiming for something more direct and exact. In particular, i'm really interested in what the shapes $H$ are. | |
| Mar 20, 2019 at 18:32 | comment | added | Alf | The minimizers are restricted classes of halfspaces. For $p \geq 2$, the blowup by an $\ell_p$ ball is no smaller than the blowup by an $\ell_2$ ball of the same radius, and this is tight for axis-aligned halfspaces. For $p < 2$, the blowup is no smaller than the blowup by an $\ell_2$ ball of radius $n^{1/2 - 1/p}\varepsilon$, and this is tight for halfpsaces with $a = \mathbf{1}$, the all-ones vector. | |
| Mar 20, 2019 at 18:18 | comment | added | Dirk | Is this answered by equivalence of norms? Probably I miss something... | |
| Mar 20, 2019 at 18:11 | history | asked | dohmatob | CC BY-SA 4.0 |