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 $beta(d,n)$?










4












$begingroup$


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?










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    4












    $begingroup$


    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?










    share|cite|improve this question









    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.







    $endgroup$














      4












      4








      4


      1



      $begingroup$


      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?










      share|cite|improve this question









      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 $mboxvol(Omega) leq 1/2$, 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






      share|cite|improve this question









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      Stefan Steinerberger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question









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      Stefan Steinerberger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




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      edited 1 hour ago







      Stefan Steinerberger













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      asked 3 hours ago









      Stefan SteinerbergerStefan Steinerberger

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      New contributor





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          1 Answer
          1






          active

          oldest

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          1












          $begingroup$

          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 $mboxvol(Omega) le frac12$.






          share|cite|improve this answer









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          • $begingroup$
            Thanks for the reference!
            $endgroup$
            – Stefan Steinerberger
            11 mins ago











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          1 Answer
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          active

          oldest

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          1 Answer
          1






          active

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          active

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          active

          oldest

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          1












          $begingroup$

          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 $mboxvol(Omega) le frac12$.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Thanks for the reference!
            $endgroup$
            – Stefan Steinerberger
            11 mins ago















          1












          $begingroup$

          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 $mboxvol(Omega) le frac12$.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Thanks for the reference!
            $endgroup$
            – Stefan Steinerberger
            11 mins ago













          1












          1








          1





          $begingroup$

          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 $mboxvol(Omega) le frac12$.






          share|cite|improve this answer









          $endgroup$



          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 $mboxvol(Omega) le frac12$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 30 mins ago









          SkeeveSkeeve

          953514




          953514











          • $begingroup$
            Thanks for the reference!
            $endgroup$
            – Stefan Steinerberger
            11 mins ago
















          • $begingroup$
            Thanks for the reference!
            $endgroup$
            – Stefan Steinerberger
            11 mins ago















          $begingroup$
          Thanks for the reference!
          $endgroup$
          – Stefan Steinerberger
          11 mins ago




          $begingroup$
          Thanks for the reference!
          $endgroup$
          – Stefan Steinerberger
          11 mins ago










          Stefan Steinerberger is a new contributor. Be nice, and check out our Code of Conduct.









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