An isoperimetric-type inequality inside a cube












8












$begingroup$


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







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    8












    $begingroup$


    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







    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.







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      8












      8








      8


      2



      $begingroup$


      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







      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 $mbox{vol}(Omega) leq 1/2$, 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






      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.











      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.









      share|cite|improve this question




      share|cite|improve this question








      edited Apr 21 at 21:43







      Stefan Steinerberger













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      asked Apr 21 at 20:11









      Stefan SteinerbergerStefan Steinerberger

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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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          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 $mbox{vol}(Omega) le frac{1}{2}$.






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          • $begingroup$
            Thanks for the reference!
            $endgroup$
            – Stefan Steinerberger
            Apr 21 at 23:26












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          active

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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 $mbox{vol}(Omega) le frac{1}{2}$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks for the reference!
            $endgroup$
            – Stefan Steinerberger
            Apr 21 at 23:26
















          6












          $begingroup$

          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 $mbox{vol}(Omega) le frac{1}{2}$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks for the reference!
            $endgroup$
            – Stefan Steinerberger
            Apr 21 at 23:26














          6












          6








          6





          $begingroup$

          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 $mbox{vol}(Omega) le frac{1}{2}$.






          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
          $|chi_Omega - mbox{vol}(Omega)|_{L^p((0,1)^d)} le C |Dchi_Omega|((0,1)^d)$, 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 $mbox{vol}(Omega) le frac{1}{2}$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Apr 21 at 23:07









          SkeeveSkeeve

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          990614












          • $begingroup$
            Thanks for the reference!
            $endgroup$
            – Stefan Steinerberger
            Apr 21 at 23:26


















          • $begingroup$
            Thanks for the reference!
            $endgroup$
            – Stefan Steinerberger
            Apr 21 at 23:26
















          $begingroup$
          Thanks for the reference!
          $endgroup$
          – Stefan Steinerberger
          Apr 21 at 23:26




          $begingroup$
          Thanks for the reference!
          $endgroup$
          – Stefan Steinerberger
          Apr 21 at 23:26










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