An isoperimetric-type inequality inside a cube Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?Name for an inequality of isoperimetric typeLevy's isoperimetric inequality for sphereStronger version of the isoperimetric inequalityIsoperimetric-like inequality for non-connected setsHypercube isoperimetric inequality for non-increasing eventsPeculiar vertex-isoperimetric inequality on the discrete torus (and generalization)Isoperimetric inequality via Crofton's formulaAn isoperimetric type of inequality in terms of Wasserstein distance/Optimal transportA cube is placed inside another cubeA question of Ahlswede and Katona: known lower bounds on $beta(d,n)$?
An isoperimetric-type inequality inside a cube
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?Name for an inequality of isoperimetric typeLevy's isoperimetric inequality for sphereStronger version of the isoperimetric inequalityIsoperimetric-like inequality for non-connected setsHypercube isoperimetric inequality for non-increasing eventsPeculiar vertex-isoperimetric inequality on the discrete torus (and generalization)Isoperimetric inequality via Crofton's formulaAn isoperimetric type of inequality in terms of Wasserstein distance/Optimal transportA cube is placed inside another cubeA question of Ahlswede and Katona: known lower bounds on $beta(d,n)$?
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I am looking for a reference for the following inequality: if $Omega subset [0,1]^d$ satisfies $mboxvol(Omega) leq 1/2$, then
$$ mathcalH^d-1left( partialOmega cap (0,1)^dright) geq c_d mboxvol(Omega)^fracd-1d,$$
where $mathcalH^d-1$ 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
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I am looking for a reference for the following inequality: if $Omega subset [0,1]^d$ satisfies $mboxvol(Omega) leq 1/2$, then
$$ mathcalH^d-1left( partialOmega cap (0,1)^dright) geq c_d mboxvol(Omega)^fracd-1d,$$
where $mathcalH^d-1$ 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
New contributor
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add a comment |
$begingroup$
I am looking for a reference for the following inequality: if $Omega subset [0,1]^d$ satisfies $mboxvol(Omega) leq 1/2$, then
$$ mathcalH^d-1left( partialOmega cap (0,1)^dright) geq c_d mboxvol(Omega)^fracd-1d,$$
where $mathcalH^d-1$ 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
New contributor
$endgroup$
I am looking for a reference for the following inequality: if $Omega subset [0,1]^d$ satisfies $mboxvol(Omega) leq 1/2$, then
$$ mathcalH^d-1left( partialOmega cap (0,1)^dright) geq c_d mboxvol(Omega)^fracd-1d,$$
where $mathcalH^d-1$ 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
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edited 3 hours ago
Stefan Steinerberger
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asked 5 hours ago
Stefan SteinerbergerStefan Steinerberger
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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
$|chi_Omega - mboxvol(Omega)|_L^p((0,1)^d) le C |Dchi_Omega|((0,1)^d)$, where $p=fracdd-1$. Here $|Dchi_Omega|((0,1)^d)=mathcalH^d-1(partial Omega cap (0,1^d))$ if $partial Omega$ is sufficiently smooth.
And
$$
|chi_Omega - mboxvol(Omega)|_p = bigl((1 - mboxvol(Omega))^p mboxvol(Omega) + mboxvol(Omega)^p (1 - mboxvol(Omega))bigr)^1/p ge frac12 mboxvol(Omega)^1/p
$$
since $mboxvol(Omega) le frac12$.
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Thanks for the reference!
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– Stefan Steinerberger
1 hour ago
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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
$|chi_Omega - mboxvol(Omega)|_L^p((0,1)^d) le C |Dchi_Omega|((0,1)^d)$, where $p=fracdd-1$. Here $|Dchi_Omega|((0,1)^d)=mathcalH^d-1(partial Omega cap (0,1^d))$ if $partial Omega$ is sufficiently smooth.
And
$$
|chi_Omega - mboxvol(Omega)|_p = bigl((1 - mboxvol(Omega))^p mboxvol(Omega) + mboxvol(Omega)^p (1 - mboxvol(Omega))bigr)^1/p ge frac12 mboxvol(Omega)^1/p
$$
since $mboxvol(Omega) le frac12$.
$endgroup$
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
1 hour 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
$|chi_Omega - mboxvol(Omega)|_L^p((0,1)^d) le C |Dchi_Omega|((0,1)^d)$, where $p=fracdd-1$. Here $|Dchi_Omega|((0,1)^d)=mathcalH^d-1(partial Omega cap (0,1^d))$ if $partial Omega$ is sufficiently smooth.
And
$$
|chi_Omega - mboxvol(Omega)|_p = bigl((1 - mboxvol(Omega))^p mboxvol(Omega) + mboxvol(Omega)^p (1 - mboxvol(Omega))bigr)^1/p ge frac12 mboxvol(Omega)^1/p
$$
since $mboxvol(Omega) le frac12$.
$endgroup$
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
1 hour 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
$|chi_Omega - mboxvol(Omega)|_L^p((0,1)^d) le C |Dchi_Omega|((0,1)^d)$, where $p=fracdd-1$. Here $|Dchi_Omega|((0,1)^d)=mathcalH^d-1(partial Omega cap (0,1^d))$ if $partial Omega$ is sufficiently smooth.
And
$$
|chi_Omega - mboxvol(Omega)|_p = bigl((1 - mboxvol(Omega))^p mboxvol(Omega) + mboxvol(Omega)^p (1 - mboxvol(Omega))bigr)^1/p ge frac12 mboxvol(Omega)^1/p
$$
since $mboxvol(Omega) le frac12$.
$endgroup$
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
$|chi_Omega - mboxvol(Omega)|_L^p((0,1)^d) le C |Dchi_Omega|((0,1)^d)$, where $p=fracdd-1$. Here $|Dchi_Omega|((0,1)^d)=mathcalH^d-1(partial Omega cap (0,1^d))$ if $partial Omega$ is sufficiently smooth.
And
$$
|chi_Omega - mboxvol(Omega)|_p = bigl((1 - mboxvol(Omega))^p mboxvol(Omega) + mboxvol(Omega)^p (1 - mboxvol(Omega))bigr)^1/p ge frac12 mboxvol(Omega)^1/p
$$
since $mboxvol(Omega) le frac12$.
answered 2 hours ago
SkeeveSkeeve
975514
975514
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
1 hour ago
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$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
1 hour ago
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
1 hour ago
$begingroup$
Thanks for the reference!
$endgroup$
– Stefan Steinerberger
1 hour 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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