Timeline for Gaussian Surface Area of Positive Semidefinite Cone
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 16, 2018 at 17:29 | answer | added | Ryan O'Donnell | timeline score: 3 | |
| Aug 25, 2017 at 8:40 | comment | added | Mateusz Kwaśnicki | I do not know the answer, although I guess it follows from Weyl's integration formula, see mathoverflow.net/a/95256/108637. | |
| Aug 24, 2017 at 23:16 | answer | added | Steve | timeline score: 3 | |
| Aug 24, 2017 at 23:04 | comment | added | Minkov | @MateuszKwaśnicki For the cone of positive semidefinite matrices, is it possible to get a closed form or order estimation for the $n-2$-dimensional measure of $A\cap \{x: |x| = 1\}$? | |
| Aug 24, 2017 at 11:49 | history | edited | Minkov | CC BY-SA 3.0 |
added 12 characters in body
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| Aug 24, 2017 at 11:49 | comment | added | Mateusz Kwaśnicki | If $A$ is a "cone" (a positively homogeneous set) with Lipschitz boundary, then $\partial A$ is also a "cone". Denote $F=\partial A\cap \partial B(0,1)$. The $n-1$-dimensional Hausdorff measure $\sigma_{n-1}$ on $\partial A$ can be written as $\sigma_{n-1}(dx)=r^{n-2}\sigma_{n-2}(dz)dr$, where $x=rz$, $r>0$, $z\in F$ and $\sigma_{n-2}$ is the $n-2$-dimensional Hausdorff measure on $F$. Now apply this to the equivalent definition of the Gaussian perimeter: $\tau(A)=\int_{\partial A}\gamma(x)\sigma_{n-1}(dx)$, where $\gamma(x)$ is the density of the Gaussian measure. | |
| Aug 24, 2017 at 11:28 | comment | added | Minkov | @MateuszKwaśnicki Sorry for the confusion. I meant the cone of positive semidefinite matrices. I will revise my statement in the question. Could you detail your claim in an answer? Thanks a lot! | |
| Aug 24, 2017 at 8:38 | comment | added | Mateusz Kwaśnicki | What do you mean by the "positive semidefinite cone"? The cone of positive semidefinite matrices? An orthant? For any cone $A$ with apex at the origin, the Gaussian perimeter is proportional to the $n-2$-dimensional measure of $A \cap \{x : |x|=1\}$, the constant of proportionality being $(2\pi)^{-n/2} \int_0^\infty r^{n-2} \exp(-r^2/2) dr = 2^{-3/2} \pi^{-n/2}\Gamma((n-1)/2)$. | |
| S Aug 24, 2017 at 5:08 | history | suggested | Steve | CC BY-SA 3.0 |
Correct a typo
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| Aug 24, 2017 at 5:08 | review | Suggested edits | |||
| S Aug 24, 2017 at 5:08 | |||||
| Aug 24, 2017 at 5:07 | history | asked | Minkov | CC BY-SA 3.0 |