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










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












    $begingroup$


    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?










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










      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






      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 1 hour ago







      Stefan Steinerberger













      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 3 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.






















          1 Answer
          1






          active

          oldest

          votes


















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









          $endgroup$














            Your Answer








            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "504"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });






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










            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f328607%2fan-isoperimetric-type-inequality-inside-a-cube%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









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









            $endgroup$


















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









              $endgroup$
















                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
                $|chi_Omega - mbox{vol}(Omega)|_{L^p((0,1)^d)} le C |Dchi_Omega|((0,1)^d)$, 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 17 mins ago









                SkeeveSkeeve

                953514




                953514






















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










                    draft saved

                    draft discarded


















                    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.
















                    Thanks for contributing an answer to MathOverflow!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f328607%2fan-isoperimetric-type-inequality-inside-a-cube%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    Gersau Kjelder | Navigasjonsmeny46°59′0″N 8°31′0″E46°59′0″N...

                    Hestehale Innhaldsliste Hestehale på kvinner | Hestehale på menn | Galleri | Sjå òg |...

                    What is the “three and three hundred thousand syndrome”?Who wrote the book Arena?What five creatures were...