Timeline for Relationship between volume and area
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 18, 2020 at 16:10 | vote | accept | Marouani | ||
| Jun 18, 2020 at 16:02 | vote | accept | Marouani | ||
| Jun 18, 2020 at 16:10 | |||||
| Jun 18, 2020 at 10:20 | comment | added | dohmatob | See my post below for a complete solution to the general problem. It's based on the co-area formula + Jensen's inequality. The particular case of $\mathbb C$ (i.e $n=1$) is a consequence of the elementary fact that a circle of radius $t$ has length (i.e Hausdorff measure) equal to $2\pi t$. | |
| Jun 18, 2020 at 10:17 | comment | added | dohmatob | OK, I've updated you question to more tightly reflect what you seem to intend. | |
| Jun 18, 2020 at 10:17 | history | edited | dohmatob | CC BY-SA 4.0 |
added 2 characters in body
|
| Jun 18, 2020 at 10:09 | history | edited | dohmatob | CC BY-SA 4.0 |
Removed confusion mu subscripts
|
| Jun 18, 2020 at 9:40 | comment | added | Marouani | You can forget the index but if we measure it is with respect to $\mu$ | |
| Jun 18, 2020 at 9:28 | comment | added | dohmatob | This still doesn't make sense to me. What is subscripted set $B_{\mathbf{\mu}}(r) = ???$ which appears in your integrals ? | |
| Jun 18, 2020 at 8:45 | comment | added | Marouani | $B(r) ={z\in\mathbb {C^n} ;|z|<r} $ | |
| Jun 18, 2020 at 8:43 | comment | added | Marouani | $Vol(B_\mu(r)) $ is the volume of the ball with respect to the measure $\mu dV$ | |
| Jun 17, 2020 at 18:46 | answer | added | dohmatob | timeline score: 1 | |
| Jun 17, 2020 at 18:17 | comment | added | dohmatob | You say "Let $\mu(z)dV$ is a measure on $\mathbb C^n$". So I guess $\mu$ is a function (namely, the density of your measure w.r.t Lebesgue). Q: What does it then mean to say "Let $B_\mu(r)$ be a ball in $\mathbb C^n$ ?". More precisely, $B_\mu(r) = ???$ | |
| Jun 17, 2020 at 11:35 | comment | added | YCor | Please don't self-vandalize your own post. You can delete it if you like, as long as it has no answer (and also undelete it later). | |
| Jun 17, 2020 at 11:35 | history | edited | YCor | CC BY-SA 4.0 |
un-vandalized post
|
| Jun 17, 2020 at 11:30 | history | edited | Marouani | CC BY-SA 4.0 |
deleted 429 characters in body
|
| Jun 16, 2020 at 21:43 | comment | added | LSpice |
I tried to proofread this (for example, don't mix text and math in an equation; write $\operatorname{Vol}(B)$ $\operatorname{Vol}(B)$, not Vol$(B)$ Vol$(B)$), but I'm not sure I got everything right. For example, you asked whether you could generate the inequality in $\mathbb C^n$, and I changed that to 'generalize'. Please feel free to revert or re-edit if I got it wrong.
|
|
| Jun 16, 2020 at 21:42 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading and TeX
|
| Jun 16, 2020 at 21:41 | review | First posts | |||
| Jun 16, 2020 at 22:52 | |||||
| Jun 16, 2020 at 21:36 | history | asked | Marouani | CC BY-SA 4.0 |