Let $E subset mathbb{R}^n$ be a convex set. Prove there exists an open set $A subset mathbb{R}^n$ equivalent...











up vote
0
down vote

favorite












Let $E subset mathbb{R}^n$ be a convex set. I need to prove there exists an open set $A subset mathbb{R}^n$ equivalent to $E$, that means such $mathcal{L}^{n} (A triangle E)= 0$, where $E triangle A = (E setminus A ) cup ( A setminus E) $.



My idea: I know that for all $epsilon >0$ there exist an open set $A supset E$ such that $mathcal{L}^{n} (A setminus E) < epsilon$, but it doesn't help because, in this way, the measure of the symmetric difference $A triangle E$ is less than $epsilon$ but it could be grater than $0$.



My second idea: If I set $A := Int(E)$ then $A$ is open and it holds $mathcal{L}^{n} (A triangle E)= mathcal{L}^{n} (E setminus A) leq mathcal{L}^{n} (partial E)$ (in the case $E$ is closed we have the equality). But in general we do NOT have $mathcal{L}^{n}( partial E) = 0$.



In both attempts I haven't used the fact that $E$ is convext, but I don't know how to use it. Does this fact imply that $mathcal{L}^{n}( partial E) = 0$? If it is true, do you know a simple proof of it?



Thank you










share|cite|improve this question


























    up vote
    0
    down vote

    favorite












    Let $E subset mathbb{R}^n$ be a convex set. I need to prove there exists an open set $A subset mathbb{R}^n$ equivalent to $E$, that means such $mathcal{L}^{n} (A triangle E)= 0$, where $E triangle A = (E setminus A ) cup ( A setminus E) $.



    My idea: I know that for all $epsilon >0$ there exist an open set $A supset E$ such that $mathcal{L}^{n} (A setminus E) < epsilon$, but it doesn't help because, in this way, the measure of the symmetric difference $A triangle E$ is less than $epsilon$ but it could be grater than $0$.



    My second idea: If I set $A := Int(E)$ then $A$ is open and it holds $mathcal{L}^{n} (A triangle E)= mathcal{L}^{n} (E setminus A) leq mathcal{L}^{n} (partial E)$ (in the case $E$ is closed we have the equality). But in general we do NOT have $mathcal{L}^{n}( partial E) = 0$.



    In both attempts I haven't used the fact that $E$ is convext, but I don't know how to use it. Does this fact imply that $mathcal{L}^{n}( partial E) = 0$? If it is true, do you know a simple proof of it?



    Thank you










    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $E subset mathbb{R}^n$ be a convex set. I need to prove there exists an open set $A subset mathbb{R}^n$ equivalent to $E$, that means such $mathcal{L}^{n} (A triangle E)= 0$, where $E triangle A = (E setminus A ) cup ( A setminus E) $.



      My idea: I know that for all $epsilon >0$ there exist an open set $A supset E$ such that $mathcal{L}^{n} (A setminus E) < epsilon$, but it doesn't help because, in this way, the measure of the symmetric difference $A triangle E$ is less than $epsilon$ but it could be grater than $0$.



      My second idea: If I set $A := Int(E)$ then $A$ is open and it holds $mathcal{L}^{n} (A triangle E)= mathcal{L}^{n} (E setminus A) leq mathcal{L}^{n} (partial E)$ (in the case $E$ is closed we have the equality). But in general we do NOT have $mathcal{L}^{n}( partial E) = 0$.



      In both attempts I haven't used the fact that $E$ is convext, but I don't know how to use it. Does this fact imply that $mathcal{L}^{n}( partial E) = 0$? If it is true, do you know a simple proof of it?



      Thank you










      share|cite|improve this question













      Let $E subset mathbb{R}^n$ be a convex set. I need to prove there exists an open set $A subset mathbb{R}^n$ equivalent to $E$, that means such $mathcal{L}^{n} (A triangle E)= 0$, where $E triangle A = (E setminus A ) cup ( A setminus E) $.



      My idea: I know that for all $epsilon >0$ there exist an open set $A supset E$ such that $mathcal{L}^{n} (A setminus E) < epsilon$, but it doesn't help because, in this way, the measure of the symmetric difference $A triangle E$ is less than $epsilon$ but it could be grater than $0$.



      My second idea: If I set $A := Int(E)$ then $A$ is open and it holds $mathcal{L}^{n} (A triangle E)= mathcal{L}^{n} (E setminus A) leq mathcal{L}^{n} (partial E)$ (in the case $E$ is closed we have the equality). But in general we do NOT have $mathcal{L}^{n}( partial E) = 0$.



      In both attempts I haven't used the fact that $E$ is convext, but I don't know how to use it. Does this fact imply that $mathcal{L}^{n}( partial E) = 0$? If it is true, do you know a simple proof of it?



      Thank you







      general-topology convex-analysis lebesgue-measure






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 22 at 14:56









      Hermione

      599




      599






















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          0
          down vote



          accepted










          Claim: if $E$ is convex, then $mathcal{L}^n(partial E) = 0$.



          Proof: Suppose otherwise. Then there is $x in partial E$ such that $limsup_{r downarrow 0} frac{|partial E cap B_r(x)|}{|B_r(x)|} = 1$. That is, $partial E$ contains most points of small balls around $x$. But by convexity, this means $E$ contains a small ball around $x$. This contradicts $x in partial E$.






          share|cite|improve this answer





















            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            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',
            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
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009238%2flet-e-subset-mathbbrn-be-a-convex-set-prove-there-exists-an-open-set-a%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








            up vote
            0
            down vote



            accepted










            Claim: if $E$ is convex, then $mathcal{L}^n(partial E) = 0$.



            Proof: Suppose otherwise. Then there is $x in partial E$ such that $limsup_{r downarrow 0} frac{|partial E cap B_r(x)|}{|B_r(x)|} = 1$. That is, $partial E$ contains most points of small balls around $x$. But by convexity, this means $E$ contains a small ball around $x$. This contradicts $x in partial E$.






            share|cite|improve this answer

























              up vote
              0
              down vote



              accepted










              Claim: if $E$ is convex, then $mathcal{L}^n(partial E) = 0$.



              Proof: Suppose otherwise. Then there is $x in partial E$ such that $limsup_{r downarrow 0} frac{|partial E cap B_r(x)|}{|B_r(x)|} = 1$. That is, $partial E$ contains most points of small balls around $x$. But by convexity, this means $E$ contains a small ball around $x$. This contradicts $x in partial E$.






              share|cite|improve this answer























                up vote
                0
                down vote



                accepted







                up vote
                0
                down vote



                accepted






                Claim: if $E$ is convex, then $mathcal{L}^n(partial E) = 0$.



                Proof: Suppose otherwise. Then there is $x in partial E$ such that $limsup_{r downarrow 0} frac{|partial E cap B_r(x)|}{|B_r(x)|} = 1$. That is, $partial E$ contains most points of small balls around $x$. But by convexity, this means $E$ contains a small ball around $x$. This contradicts $x in partial E$.






                share|cite|improve this answer












                Claim: if $E$ is convex, then $mathcal{L}^n(partial E) = 0$.



                Proof: Suppose otherwise. Then there is $x in partial E$ such that $limsup_{r downarrow 0} frac{|partial E cap B_r(x)|}{|B_r(x)|} = 1$. That is, $partial E$ contains most points of small balls around $x$. But by convexity, this means $E$ contains a small ball around $x$. This contradicts $x in partial E$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 22 at 15:05









                mathworker21

                8,2761827




                8,2761827






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • 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.





                    Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                    Please pay close attention to the following guidance:


                    • 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.


                    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%2fmath.stackexchange.com%2fquestions%2f3009238%2flet-e-subset-mathbbrn-be-a-convex-set-prove-there-exists-an-open-set-a%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

                    Probability when a professor distributes a quiz and homework assignment to a class of n students.

                    Aardman Animations

                    Are they similar matrix