Distance to closed and convex sets that have intersection $C subseteq D rightarrow$ $ d_D(x^*) leq d_C(x^*) $












0














Let $C subseteq mathbb{R}^n$ be a closed convex set, and $x^* in C^c$ (not in $C$ and its closure).



Define the Euclidean distance from $x^*$ to $C$ as $d_C(x^*):=min_{z in C}|z -x^*|_2$.



Let $D$ be a closed convex set containing $C$, i.e., $C subseteq D$.



Show that
$$
d_D(x^*) leq d_C(x^*)
$$



I do not know how to use $C subseteq D$ together with taking minimum.






optimization convex-optimization projective-geometry







share|cite|improve this question






















  • Assume the distance is strictly smaller, use minimum property and that every element in $C$ is an element in $D$. That leads to a contradiction.
    – B.Swan
    Nov 25 at 7:57


















0














Let $C subseteq mathbb{R}^n$ be a closed convex set, and $x^* in C^c$ (not in $C$ and its closure).



Define the Euclidean distance from $x^*$ to $C$ as $d_C(x^*):=min_{z in C}|z -x^*|_2$.



Let $D$ be a closed convex set containing $C$, i.e., $C subseteq D$.



Show that
$$
d_D(x^*) leq d_C(x^*)
$$



I do not know how to use $C subseteq D$ together with taking minimum.






optimization convex-optimization projective-geometry







share|cite|improve this question






















  • Assume the distance is strictly smaller, use minimum property and that every element in $C$ is an element in $D$. That leads to a contradiction.
    – B.Swan
    Nov 25 at 7:57
















0












0








0







Let $C subseteq mathbb{R}^n$ be a closed convex set, and $x^* in C^c$ (not in $C$ and its closure).



Define the Euclidean distance from $x^*$ to $C$ as $d_C(x^*):=min_{z in C}|z -x^*|_2$.



Let $D$ be a closed convex set containing $C$, i.e., $C subseteq D$.



Show that
$$
d_D(x^*) leq d_C(x^*)
$$



I do not know how to use $C subseteq D$ together with taking minimum.






optimization convex-optimization projective-geometry







share|cite|improve this question













Let $C subseteq mathbb{R}^n$ be a closed convex set, and $x^* in C^c$ (not in $C$ and its closure).



Define the Euclidean distance from $x^*$ to $C$ as $d_C(x^*):=min_{z in C}|z -x^*|_2$.



Let $D$ be a closed convex set containing $C$, i.e., $C subseteq D$.



Show that
$$
d_D(x^*) leq d_C(x^*)
$$



I do not know how to use $C subseteq D$ together with taking minimum.






optimization convex-optimization projective-geometry




optimization convex-optimization projective-geometry






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 25 at 7:53









Saeed

585110




585110












  • Assume the distance is strictly smaller, use minimum property and that every element in $C$ is an element in $D$. That leads to a contradiction.
    – B.Swan
    Nov 25 at 7:57




















  • Assume the distance is strictly smaller, use minimum property and that every element in $C$ is an element in $D$. That leads to a contradiction.
    – B.Swan
    Nov 25 at 7:57


















Assume the distance is strictly smaller, use minimum property and that every element in $C$ is an element in $D$. That leads to a contradiction.
– B.Swan
Nov 25 at 7:57






Assume the distance is strictly smaller, use minimum property and that every element in $C$ is an element in $D$. That leads to a contradiction.
– B.Swan
Nov 25 at 7:57












2 Answers
2






active

oldest

votes


















1














Since $C subseteq D$, the minimum over $C$ can only be greater or equal to the minimum over $D$, so



$$d_D(x^*)=min_{z in D}|z -x^*|_2 leq min_{z in C}|z -x^*|_2= d_D(x^*)$$






share|cite|improve this answer





























    0














    for $forall z in D$ and $yin C$, we have
    begin{align}
    d_D(x^*) &= min_{z in D} |z-x^*|_2 \
    &= min_{yin C} |(z_{best} -y)+ (y-x^*)|_2 \
    &leq min_{y in C}|z_{best}-y|_2 + min_{y in C}|y-x^*|_2 \
    & leq min_{yin C}|y-x^*|_2 \
    &=d_C(x^*)
    end{align}




    Answer above has mistakes. while, maybe we can try a new way to solve this.



    Notice that for any element $y in C$, because $C subseteq D$, so that $y in D$, and another element $z_{best} in D$, satisfy $d_D(x^*)=min_{z in D} |z-x^*|_2=|z_{best}-x^*|_2$, from this definition, it is quite clear to see
    $$ |y-x^*|_2 geq |z_{best}-x^*|_2, forall y in C$$
    which results in
    $$d_C(x^*) = min_{yin C}|y-x^*|_2 geq |z_{best}-x^*|_2 = d_D(x^*)$$






    share|cite|improve this answer























    • How do you know when you ignore $min_{y in C}|z_{best}-y|_2$ inequality still holds?
      – Saeed
      Nov 25 at 18:59










    • You are right! And I post a new solution under the original one. I hope it can help you.
      – Caldera
      Nov 26 at 7:50











    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',
    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
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3012558%2fdistance-to-closed-and-convex-sets-that-have-intersection-c-subseteq-d-righta%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1














    Since $C subseteq D$, the minimum over $C$ can only be greater or equal to the minimum over $D$, so



    $$d_D(x^*)=min_{z in D}|z -x^*|_2 leq min_{z in C}|z -x^*|_2= d_D(x^*)$$






    share|cite|improve this answer


























      1














      Since $C subseteq D$, the minimum over $C$ can only be greater or equal to the minimum over $D$, so



      $$d_D(x^*)=min_{z in D}|z -x^*|_2 leq min_{z in C}|z -x^*|_2= d_D(x^*)$$






      share|cite|improve this answer
























        1












        1








        1






        Since $C subseteq D$, the minimum over $C$ can only be greater or equal to the minimum over $D$, so



        $$d_D(x^*)=min_{z in D}|z -x^*|_2 leq min_{z in C}|z -x^*|_2= d_D(x^*)$$






        share|cite|improve this answer












        Since $C subseteq D$, the minimum over $C$ can only be greater or equal to the minimum over $D$, so



        $$d_D(x^*)=min_{z in D}|z -x^*|_2 leq min_{z in C}|z -x^*|_2= d_D(x^*)$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 25 at 8:05









        B.Swan

        1,0011719




        1,0011719























            0














            for $forall z in D$ and $yin C$, we have
            begin{align}
            d_D(x^*) &= min_{z in D} |z-x^*|_2 \
            &= min_{yin C} |(z_{best} -y)+ (y-x^*)|_2 \
            &leq min_{y in C}|z_{best}-y|_2 + min_{y in C}|y-x^*|_2 \
            & leq min_{yin C}|y-x^*|_2 \
            &=d_C(x^*)
            end{align}




            Answer above has mistakes. while, maybe we can try a new way to solve this.



            Notice that for any element $y in C$, because $C subseteq D$, so that $y in D$, and another element $z_{best} in D$, satisfy $d_D(x^*)=min_{z in D} |z-x^*|_2=|z_{best}-x^*|_2$, from this definition, it is quite clear to see
            $$ |y-x^*|_2 geq |z_{best}-x^*|_2, forall y in C$$
            which results in
            $$d_C(x^*) = min_{yin C}|y-x^*|_2 geq |z_{best}-x^*|_2 = d_D(x^*)$$






            share|cite|improve this answer























            • How do you know when you ignore $min_{y in C}|z_{best}-y|_2$ inequality still holds?
              – Saeed
              Nov 25 at 18:59










            • You are right! And I post a new solution under the original one. I hope it can help you.
              – Caldera
              Nov 26 at 7:50
















            0














            for $forall z in D$ and $yin C$, we have
            begin{align}
            d_D(x^*) &= min_{z in D} |z-x^*|_2 \
            &= min_{yin C} |(z_{best} -y)+ (y-x^*)|_2 \
            &leq min_{y in C}|z_{best}-y|_2 + min_{y in C}|y-x^*|_2 \
            & leq min_{yin C}|y-x^*|_2 \
            &=d_C(x^*)
            end{align}




            Answer above has mistakes. while, maybe we can try a new way to solve this.



            Notice that for any element $y in C$, because $C subseteq D$, so that $y in D$, and another element $z_{best} in D$, satisfy $d_D(x^*)=min_{z in D} |z-x^*|_2=|z_{best}-x^*|_2$, from this definition, it is quite clear to see
            $$ |y-x^*|_2 geq |z_{best}-x^*|_2, forall y in C$$
            which results in
            $$d_C(x^*) = min_{yin C}|y-x^*|_2 geq |z_{best}-x^*|_2 = d_D(x^*)$$






            share|cite|improve this answer























            • How do you know when you ignore $min_{y in C}|z_{best}-y|_2$ inequality still holds?
              – Saeed
              Nov 25 at 18:59










            • You are right! And I post a new solution under the original one. I hope it can help you.
              – Caldera
              Nov 26 at 7:50














            0












            0








            0






            for $forall z in D$ and $yin C$, we have
            begin{align}
            d_D(x^*) &= min_{z in D} |z-x^*|_2 \
            &= min_{yin C} |(z_{best} -y)+ (y-x^*)|_2 \
            &leq min_{y in C}|z_{best}-y|_2 + min_{y in C}|y-x^*|_2 \
            & leq min_{yin C}|y-x^*|_2 \
            &=d_C(x^*)
            end{align}




            Answer above has mistakes. while, maybe we can try a new way to solve this.



            Notice that for any element $y in C$, because $C subseteq D$, so that $y in D$, and another element $z_{best} in D$, satisfy $d_D(x^*)=min_{z in D} |z-x^*|_2=|z_{best}-x^*|_2$, from this definition, it is quite clear to see
            $$ |y-x^*|_2 geq |z_{best}-x^*|_2, forall y in C$$
            which results in
            $$d_C(x^*) = min_{yin C}|y-x^*|_2 geq |z_{best}-x^*|_2 = d_D(x^*)$$






            share|cite|improve this answer














            for $forall z in D$ and $yin C$, we have
            begin{align}
            d_D(x^*) &= min_{z in D} |z-x^*|_2 \
            &= min_{yin C} |(z_{best} -y)+ (y-x^*)|_2 \
            &leq min_{y in C}|z_{best}-y|_2 + min_{y in C}|y-x^*|_2 \
            & leq min_{yin C}|y-x^*|_2 \
            &=d_C(x^*)
            end{align}




            Answer above has mistakes. while, maybe we can try a new way to solve this.



            Notice that for any element $y in C$, because $C subseteq D$, so that $y in D$, and another element $z_{best} in D$, satisfy $d_D(x^*)=min_{z in D} |z-x^*|_2=|z_{best}-x^*|_2$, from this definition, it is quite clear to see
            $$ |y-x^*|_2 geq |z_{best}-x^*|_2, forall y in C$$
            which results in
            $$d_C(x^*) = min_{yin C}|y-x^*|_2 geq |z_{best}-x^*|_2 = d_D(x^*)$$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Nov 26 at 7:48

























            answered Nov 25 at 8:17









            Caldera

            11




            11












            • How do you know when you ignore $min_{y in C}|z_{best}-y|_2$ inequality still holds?
              – Saeed
              Nov 25 at 18:59










            • You are right! And I post a new solution under the original one. I hope it can help you.
              – Caldera
              Nov 26 at 7:50


















            • How do you know when you ignore $min_{y in C}|z_{best}-y|_2$ inequality still holds?
              – Saeed
              Nov 25 at 18:59










            • You are right! And I post a new solution under the original one. I hope it can help you.
              – Caldera
              Nov 26 at 7:50
















            How do you know when you ignore $min_{y in C}|z_{best}-y|_2$ inequality still holds?
            – Saeed
            Nov 25 at 18:59




            How do you know when you ignore $min_{y in C}|z_{best}-y|_2$ inequality still holds?
            – Saeed
            Nov 25 at 18:59












            You are right! And I post a new solution under the original one. I hope it can help you.
            – Caldera
            Nov 26 at 7:50




            You are right! And I post a new solution under the original one. I hope it can help you.
            – Caldera
            Nov 26 at 7:50


















            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%2f3012558%2fdistance-to-closed-and-convex-sets-that-have-intersection-c-subseteq-d-righta%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.

            Image
            0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
            Read more

            Aardman Animations

            Image
            Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
            Read more

            Are they similar matrix

            Image
            0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
            Read more