An isoperimetric-type inequality inside a cube Planned maintenance scheduled April 23, 2019 at...
An isoperimetric-type inequality inside a cube
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)
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$begingroup$
I am looking for a reference for the following inequality: if $Omega subset [0,1]^d$ satisfies $mbox{vol}(Omega) leq 1/2$, then
$$ mathcal{H}^{d-1}left( partialOmega cap (0,1)^dright) geq c_d mbox{vol}(Omega)^{frac{d-1}{d}},$$
where $mathcal{H}^{d-1}$ is the $(d-1)-$dimensional Hausdorff measure and $c_d > 0$ is a universal constant depending only on $d$.
This is a variation of the classical isoperimetric inequality with the interesting addition that surface 'on the boundary of the cube' does not count. This seems like it should be known. A discrete version of this inequality (for subsets of the grid graph) was proven by Bollobas and Leader (Edge-isoperimetric inequalities in the grid, Combinatorica 1991) and it seems there is a wealth of information for the discrete case. Has anybody seen the continuous case stated somewhere?
reference-request mg.metric-geometry geometric-measure-theory isoperimetric-problems
asked 4 hours ago
Stefan SteinerbergerStefan Steinerberger
233
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Stefan Steinerberger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$
add a comment |
$begingroup$
I am looking for a reference for the following inequality: if $Omega subset [0,1]^d$ satisfies $mbox{vol}(Omega) leq 1/2$, then
$$ mathcal{H}^{d-1}left( partialOmega cap (0,1)^dright) geq c_d mbox{vol}(Omega)^{frac{d-1}{d}},$$
where $mathcal{H}^{d-1}$ is the $(d-1)-$dimensional Hausdorff measure and $c_d > 0$ is a universal constant depending only on $d$.
This is a variation of the classical isoperimetric inequality with the interesting addition that surface 'on the boundary of the cube' does not count. This seems like it should be known. A discrete version of this inequality (for subsets of the grid graph) was proven by Bollobas and Leader (Edge-isoperimetric inequalities in the grid, Combinatorica 1991) and it seems there is a wealth of information for the discrete case. Has anybody seen the continuous case stated somewhere?
reference-request mg.metric-geometry geometric-measure-theory isoperimetric-problems
asked 4 hours ago
Stefan SteinerbergerStefan Steinerberger
233
New contributor
Stefan Steinerberger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I am looking for a reference for the following inequality: if $Omega subset [0,1]^d$ satisfies $mbox{vol}(Omega) leq 1/2$, then
$$ mathcal{H}^{d-1}left( partialOmega cap (0,1)^dright) geq c_d mbox{vol}(Omega)^{frac{d-1}{d}},$$
where $mathcal{H}^{d-1}$ is the $(d-1)-$dimensional Hausdorff measure and $c_d > 0$ is a universal constant depending only on $d$.
This is a variation of the classical isoperimetric inequality with the interesting addition that surface 'on the boundary of the cube' does not count. This seems like it should be known. A discrete version of this inequality (for subsets of the grid graph) was proven by Bollobas and Leader (Edge-isoperimetric inequalities in the grid, Combinatorica 1991) and it seems there is a wealth of information for the discrete case. Has anybody seen the continuous case stated somewhere?
reference-request mg.metric-geometry geometric-measure-theory isoperimetric-problems
asked 4 hours ago
Stefan SteinerbergerStefan Steinerberger
233
New contributor
Stefan Steinerberger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I am looking for a reference for the following inequality: if $Omega subset [0,1]^d$ satisfies $mbox{vol}(Omega) leq 1/2$, then
$$ mathcal{H}^{d-1}left( partialOmega cap (0,1)^dright) geq c_d mbox{vol}(Omega)^{frac{d-1}{d}},$$
where $mathcal{H}^{d-1}$ is the $(d-1)-$dimensional Hausdorff measure and $c_d > 0$ is a universal constant depending only on $d$.
This is a variation of the classical isoperimetric inequality with the interesting addition that surface 'on the boundary of the cube' does not count. This seems like it should be known. A discrete version of this inequality (for subsets of the grid graph) was proven by Bollobas and Leader (Edge-isoperimetric inequalities in the grid, Combinatorica 1991) and it seems there is a wealth of information for the discrete case. Has anybody seen the continuous case stated somewhere?
reference-request mg.metric-geometry geometric-measure-theory isoperimetric-problems
reference-request mg.metric-geometry geometric-measure-theory isoperimetric-problems
asked 4 hours ago
Stefan SteinerbergerStefan Steinerberger
233
New contributor
Stefan Steinerberger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Stefan SteinerbergerStefan Steinerberger
233
New contributor
Stefan Steinerberger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 hours ago
Stefan Steinerberger
asked 4 hours ago
Stefan SteinerbergerStefan Steinerberger
233
New contributor
Stefan Steinerberger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Stefan SteinerbergerStefan Steinerberger
233
asked 4 hours ago
Stefan SteinerbergerStefan Steinerberger
233
233
New contributor
Stefan Steinerberger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Stefan Steinerberger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Stefan Steinerberger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
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$begingroup$
This result is known as the relative isoperimetric inequality, see e.g. Functions of Bounded Variation by L. Ambrosio, N. Fusco and D. Pallara (2000), Eq. (3.43).
It follows from Poincare inequality (see e.g. Eq. (3.41) in the cited book) applied to $chi_Omega$ (the indicator of the set $Omega$). Indeed, by Poincare inequality it holds
$|chi_Omega - mbox{vol}(Omega)|_{L^p((0,1)^d)} le C |Dchi_Omega|((0,1)^d)$, where $p=frac{d}{d-1}$. Here $|Dchi_Omega|((0,1)^d)=mathcal{H}^{d-1}(partial Omega cap (0,1^d))$ if $partial Omega$ is sufficiently smooth.
And
$$
|chi_Omega - mbox{vol}(Omega)|_p = bigl((1 - mbox{vol}(Omega))^p mbox{vol}(Omega) + mbox{vol}(Omega)^p (1 - mbox{vol}(Omega))bigr)^{1/p} ge frac{1}{2} mbox{vol}(Omega)^{1/p}
$$
since $mbox{vol}(Omega) le frac{1}{2}$.
answered 1 hour ago
SkeeveSkeeve
955514
$endgroup$
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
52 mins ago
add a comment |
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1 Answer
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1 Answer
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active
oldest
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votes
$begingroup$
This result is known as the relative isoperimetric inequality, see e.g. Functions of Bounded Variation by L. Ambrosio, N. Fusco and D. Pallara (2000), Eq. (3.43).
It follows from Poincare inequality (see e.g. Eq. (3.41) in the cited book) applied to $chi_Omega$ (the indicator of the set $Omega$). Indeed, by Poincare inequality it holds
$|chi_Omega - mbox{vol}(Omega)|_{L^p((0,1)^d)} le C |Dchi_Omega|((0,1)^d)$, where $p=frac{d}{d-1}$. Here $|Dchi_Omega|((0,1)^d)=mathcal{H}^{d-1}(partial Omega cap (0,1^d))$ if $partial Omega$ is sufficiently smooth.
And
$$
|chi_Omega - mbox{vol}(Omega)|_p = bigl((1 - mbox{vol}(Omega))^p mbox{vol}(Omega) + mbox{vol}(Omega)^p (1 - mbox{vol}(Omega))bigr)^{1/p} ge frac{1}{2} mbox{vol}(Omega)^{1/p}
$$
since $mbox{vol}(Omega) le frac{1}{2}$.
answered 1 hour ago
SkeeveSkeeve
955514
$endgroup$
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
52 mins ago
add a comment |
$begingroup$
This result is known as the relative isoperimetric inequality, see e.g. Functions of Bounded Variation by L. Ambrosio, N. Fusco and D. Pallara (2000), Eq. (3.43).
It follows from Poincare inequality (see e.g. Eq. (3.41) in the cited book) applied to $chi_Omega$ (the indicator of the set $Omega$). Indeed, by Poincare inequality it holds
$|chi_Omega - mbox{vol}(Omega)|_{L^p((0,1)^d)} le C |Dchi_Omega|((0,1)^d)$, where $p=frac{d}{d-1}$. Here $|Dchi_Omega|((0,1)^d)=mathcal{H}^{d-1}(partial Omega cap (0,1^d))$ if $partial Omega$ is sufficiently smooth.
And
$$
|chi_Omega - mbox{vol}(Omega)|_p = bigl((1 - mbox{vol}(Omega))^p mbox{vol}(Omega) + mbox{vol}(Omega)^p (1 - mbox{vol}(Omega))bigr)^{1/p} ge frac{1}{2} mbox{vol}(Omega)^{1/p}
$$
since $mbox{vol}(Omega) le frac{1}{2}$.
answered 1 hour ago
SkeeveSkeeve
955514
$endgroup$
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
52 mins ago
add a comment |
$begingroup$
This result is known as the relative isoperimetric inequality, see e.g. Functions of Bounded Variation by L. Ambrosio, N. Fusco and D. Pallara (2000), Eq. (3.43).
It follows from Poincare inequality (see e.g. Eq. (3.41) in the cited book) applied to $chi_Omega$ (the indicator of the set $Omega$). Indeed, by Poincare inequality it holds
$|chi_Omega - mbox{vol}(Omega)|_{L^p((0,1)^d)} le C |Dchi_Omega|((0,1)^d)$, where $p=frac{d}{d-1}$. Here $|Dchi_Omega|((0,1)^d)=mathcal{H}^{d-1}(partial Omega cap (0,1^d))$ if $partial Omega$ is sufficiently smooth.
And
$$
|chi_Omega - mbox{vol}(Omega)|_p = bigl((1 - mbox{vol}(Omega))^p mbox{vol}(Omega) + mbox{vol}(Omega)^p (1 - mbox{vol}(Omega))bigr)^{1/p} ge frac{1}{2} mbox{vol}(Omega)^{1/p}
$$
since $mbox{vol}(Omega) le frac{1}{2}$.
answered 1 hour ago
SkeeveSkeeve
955514
$endgroup$
This result is known as the relative isoperimetric inequality, see e.g. Functions of Bounded Variation by L. Ambrosio, N. Fusco and D. Pallara (2000), Eq. (3.43).
It follows from Poincare inequality (see e.g. Eq. (3.41) in the cited book) applied to $chi_Omega$ (the indicator of the set $Omega$). Indeed, by Poincare inequality it holds
$|chi_Omega - mbox{vol}(Omega)|_{L^p((0,1)^d)} le C |Dchi_Omega|((0,1)^d)$, where $p=frac{d}{d-1}$. Here $|Dchi_Omega|((0,1)^d)=mathcal{H}^{d-1}(partial Omega cap (0,1^d))$ if $partial Omega$ is sufficiently smooth.
And
$$
|chi_Omega - mbox{vol}(Omega)|_p = bigl((1 - mbox{vol}(Omega))^p mbox{vol}(Omega) + mbox{vol}(Omega)^p (1 - mbox{vol}(Omega))bigr)^{1/p} ge frac{1}{2} mbox{vol}(Omega)^{1/p}
$$
since $mbox{vol}(Omega) le frac{1}{2}$.
answered 1 hour ago
SkeeveSkeeve
955514
answered 1 hour ago
SkeeveSkeeve
955514
answered 1 hour ago
SkeeveSkeeve
955514
answered 1 hour ago
SkeeveSkeeve
955514
955514
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
52 mins ago
add a comment |
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
52 mins ago
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
52 mins ago
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
52 mins ago
add a comment |
Stefan Steinerberger is a new contributor. Be nice, and check out our Code of Conduct.
Stefan Steinerberger is a new contributor. Be nice, and check out our Code of Conduct.
Stefan Steinerberger is a new contributor. Be nice, and check out our Code of Conduct.
Stefan Steinerberger is a new contributor. Be nice, and check out our Code of Conduct.
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