Vector space is to manifold as convex cone is to?












4












$begingroup$


In convex analysis, a convex cone can be viewed as being like a one-sided version of a subspace. (And the polar cone is analogous to the orthogonal complement. It's a nice analogy.)



A smooth manifold has a tangent space at every point. Is there an analogous type of mathematical structure which is sort of like a smooth manifold, but which has a convex cone at every point? (Or can be approximated in some sense by a convex cone at every point?)



If so then what is the name of this type of object?






convex-analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    Timelike spaces?
    $endgroup$
    – mma
    May 31 '17 at 20:13
















4












$begingroup$


In convex analysis, a convex cone can be viewed as being like a one-sided version of a subspace. (And the polar cone is analogous to the orthogonal complement. It's a nice analogy.)



A smooth manifold has a tangent space at every point. Is there an analogous type of mathematical structure which is sort of like a smooth manifold, but which has a convex cone at every point? (Or can be approximated in some sense by a convex cone at every point?)



If so then what is the name of this type of object?






convex-analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    Timelike spaces?
    $endgroup$
    – mma
    May 31 '17 at 20:13














4












4








4


3



$begingroup$


In convex analysis, a convex cone can be viewed as being like a one-sided version of a subspace. (And the polar cone is analogous to the orthogonal complement. It's a nice analogy.)



A smooth manifold has a tangent space at every point. Is there an analogous type of mathematical structure which is sort of like a smooth manifold, but which has a convex cone at every point? (Or can be approximated in some sense by a convex cone at every point?)



If so then what is the name of this type of object?






convex-analysis







share|cite|improve this question











$endgroup$




In convex analysis, a convex cone can be viewed as being like a one-sided version of a subspace. (And the polar cone is analogous to the orthogonal complement. It's a nice analogy.)



A smooth manifold has a tangent space at every point. Is there an analogous type of mathematical structure which is sort of like a smooth manifold, but which has a convex cone at every point? (Or can be approximated in some sense by a convex cone at every point?)



If so then what is the name of this type of object?






convex-analysis




convex-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 23 '17 at 19:13







eternalGoldenBraid

















asked Jan 23 '17 at 10:36









eternalGoldenBraideternalGoldenBraid

727314




727314












  • $begingroup$
    Timelike spaces?
    $endgroup$
    – mma
    May 31 '17 at 20:13


















  • $begingroup$
    Timelike spaces?
    $endgroup$
    – mma
    May 31 '17 at 20:13
















$begingroup$
Timelike spaces?
$endgroup$
– mma
May 31 '17 at 20:13




$begingroup$
Timelike spaces?
$endgroup$
– mma
May 31 '17 at 20:13










1 Answer
1






active

oldest

votes


















1












$begingroup$

My opinion is that the answer is contained in your first words: convex sets.




  • Convex sets have tangent and normal cones at any point.

  • The tangent cone to a convex set is a good local approximation $quad T_C(x) := text{cl } mathbb{R}_+(C-x)$

  • These tangent and normal cones are convex, and in duality by polarity.


This being said, you could argue that it is not really satisfying because "convex sets" and "manifolds" are two separate class of sets (even though there is some overlap between the two).
Then, you might want to go into nonsmooth variational analysis, a field where people study nonconvex nonsmooth sets, their first-order approximations, etc.



For instance there is the interesting class of prox-regular sets (for simplicity I consider the ambient space to be $mathbb{R}^N$):




  • A set $S$ is said to be prox regular if the metric projection $x mapsto text{proj}_C(x)$ is uniquely defined in a neighbourhood of $S$.

  • Examples of prox-regular sets are convex sets and manifolds of class $C^2$ (but not only).

  • There are various possible notions of tangent/normal cones for these sets in the litterature. The interesting point is that they all reduce to the classic tangent/normal cones when $S$ is convex, or to the classic tangent/normal spaces when $S$ is a manifold. Generally people consider the so-called Limiting normal cone [1, Theorem 1.6]
    $$ N^{L}_S(x) := underset{y to x}{text{Limsup}} mathbb{R}_+(y - text{proj}_S(y)), $$
    and take its polar as a tangent cone: $T^L_S(x) := N^L_S(x)^0$.

  • By definition, the Limiting tangent space is convex, but unfortunately, the Limiting normal cone is not convex in general. We also lose a perfect duality between tangent and normal cones : $N^L_S(x) subsetneq T^L_S(x)^0$.


[1] Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation I: Basic theory (Vol. 330). Springer Science & Business Media.






share|cite|improve this answer









$endgroup$













    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%2f2110094%2fvector-space-is-to-manifold-as-convex-cone-is-to%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












    $begingroup$

    My opinion is that the answer is contained in your first words: convex sets.




    • Convex sets have tangent and normal cones at any point.

    • The tangent cone to a convex set is a good local approximation $quad T_C(x) := text{cl } mathbb{R}_+(C-x)$

    • These tangent and normal cones are convex, and in duality by polarity.


    This being said, you could argue that it is not really satisfying because "convex sets" and "manifolds" are two separate class of sets (even though there is some overlap between the two).
    Then, you might want to go into nonsmooth variational analysis, a field where people study nonconvex nonsmooth sets, their first-order approximations, etc.



    For instance there is the interesting class of prox-regular sets (for simplicity I consider the ambient space to be $mathbb{R}^N$):




    • A set $S$ is said to be prox regular if the metric projection $x mapsto text{proj}_C(x)$ is uniquely defined in a neighbourhood of $S$.

    • Examples of prox-regular sets are convex sets and manifolds of class $C^2$ (but not only).

    • There are various possible notions of tangent/normal cones for these sets in the litterature. The interesting point is that they all reduce to the classic tangent/normal cones when $S$ is convex, or to the classic tangent/normal spaces when $S$ is a manifold. Generally people consider the so-called Limiting normal cone [1, Theorem 1.6]
      $$ N^{L}_S(x) := underset{y to x}{text{Limsup}} mathbb{R}_+(y - text{proj}_S(y)), $$
      and take its polar as a tangent cone: $T^L_S(x) := N^L_S(x)^0$.

    • By definition, the Limiting tangent space is convex, but unfortunately, the Limiting normal cone is not convex in general. We also lose a perfect duality between tangent and normal cones : $N^L_S(x) subsetneq T^L_S(x)^0$.


    [1] Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation I: Basic theory (Vol. 330). Springer Science & Business Media.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      My opinion is that the answer is contained in your first words: convex sets.




      • Convex sets have tangent and normal cones at any point.

      • The tangent cone to a convex set is a good local approximation $quad T_C(x) := text{cl } mathbb{R}_+(C-x)$

      • These tangent and normal cones are convex, and in duality by polarity.


      This being said, you could argue that it is not really satisfying because "convex sets" and "manifolds" are two separate class of sets (even though there is some overlap between the two).
      Then, you might want to go into nonsmooth variational analysis, a field where people study nonconvex nonsmooth sets, their first-order approximations, etc.



      For instance there is the interesting class of prox-regular sets (for simplicity I consider the ambient space to be $mathbb{R}^N$):




      • A set $S$ is said to be prox regular if the metric projection $x mapsto text{proj}_C(x)$ is uniquely defined in a neighbourhood of $S$.

      • Examples of prox-regular sets are convex sets and manifolds of class $C^2$ (but not only).

      • There are various possible notions of tangent/normal cones for these sets in the litterature. The interesting point is that they all reduce to the classic tangent/normal cones when $S$ is convex, or to the classic tangent/normal spaces when $S$ is a manifold. Generally people consider the so-called Limiting normal cone [1, Theorem 1.6]
        $$ N^{L}_S(x) := underset{y to x}{text{Limsup}} mathbb{R}_+(y - text{proj}_S(y)), $$
        and take its polar as a tangent cone: $T^L_S(x) := N^L_S(x)^0$.

      • By definition, the Limiting tangent space is convex, but unfortunately, the Limiting normal cone is not convex in general. We also lose a perfect duality between tangent and normal cones : $N^L_S(x) subsetneq T^L_S(x)^0$.


      [1] Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation I: Basic theory (Vol. 330). Springer Science & Business Media.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        My opinion is that the answer is contained in your first words: convex sets.




        • Convex sets have tangent and normal cones at any point.

        • The tangent cone to a convex set is a good local approximation $quad T_C(x) := text{cl } mathbb{R}_+(C-x)$

        • These tangent and normal cones are convex, and in duality by polarity.


        This being said, you could argue that it is not really satisfying because "convex sets" and "manifolds" are two separate class of sets (even though there is some overlap between the two).
        Then, you might want to go into nonsmooth variational analysis, a field where people study nonconvex nonsmooth sets, their first-order approximations, etc.



        For instance there is the interesting class of prox-regular sets (for simplicity I consider the ambient space to be $mathbb{R}^N$):




        • A set $S$ is said to be prox regular if the metric projection $x mapsto text{proj}_C(x)$ is uniquely defined in a neighbourhood of $S$.

        • Examples of prox-regular sets are convex sets and manifolds of class $C^2$ (but not only).

        • There are various possible notions of tangent/normal cones for these sets in the litterature. The interesting point is that they all reduce to the classic tangent/normal cones when $S$ is convex, or to the classic tangent/normal spaces when $S$ is a manifold. Generally people consider the so-called Limiting normal cone [1, Theorem 1.6]
          $$ N^{L}_S(x) := underset{y to x}{text{Limsup}} mathbb{R}_+(y - text{proj}_S(y)), $$
          and take its polar as a tangent cone: $T^L_S(x) := N^L_S(x)^0$.

        • By definition, the Limiting tangent space is convex, but unfortunately, the Limiting normal cone is not convex in general. We also lose a perfect duality between tangent and normal cones : $N^L_S(x) subsetneq T^L_S(x)^0$.


        [1] Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation I: Basic theory (Vol. 330). Springer Science & Business Media.






        share|cite|improve this answer









        $endgroup$



        My opinion is that the answer is contained in your first words: convex sets.




        • Convex sets have tangent and normal cones at any point.

        • The tangent cone to a convex set is a good local approximation $quad T_C(x) := text{cl } mathbb{R}_+(C-x)$

        • These tangent and normal cones are convex, and in duality by polarity.


        This being said, you could argue that it is not really satisfying because "convex sets" and "manifolds" are two separate class of sets (even though there is some overlap between the two).
        Then, you might want to go into nonsmooth variational analysis, a field where people study nonconvex nonsmooth sets, their first-order approximations, etc.



        For instance there is the interesting class of prox-regular sets (for simplicity I consider the ambient space to be $mathbb{R}^N$):




        • A set $S$ is said to be prox regular if the metric projection $x mapsto text{proj}_C(x)$ is uniquely defined in a neighbourhood of $S$.

        • Examples of prox-regular sets are convex sets and manifolds of class $C^2$ (but not only).

        • There are various possible notions of tangent/normal cones for these sets in the litterature. The interesting point is that they all reduce to the classic tangent/normal cones when $S$ is convex, or to the classic tangent/normal spaces when $S$ is a manifold. Generally people consider the so-called Limiting normal cone [1, Theorem 1.6]
          $$ N^{L}_S(x) := underset{y to x}{text{Limsup}} mathbb{R}_+(y - text{proj}_S(y)), $$
          and take its polar as a tangent cone: $T^L_S(x) := N^L_S(x)^0$.

        • By definition, the Limiting tangent space is convex, but unfortunately, the Limiting normal cone is not convex in general. We also lose a perfect duality between tangent and normal cones : $N^L_S(x) subsetneq T^L_S(x)^0$.


        [1] Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation I: Basic theory (Vol. 330). Springer Science & Business Media.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 4 '18 at 17:15









        Guillaume GarrigosGuillaume Garrigos

        1619




        1619






























            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.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2110094%2fvector-space-is-to-manifold-as-convex-cone-is-to%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

            Plaza Victoria

            Image
            Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l
            Read more

            In PowerPoint, is there a keyboard shortcut for bulleted / numbered list?

            Image
            .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 2 OneNote has two useful keyboard shortcuts for bulleted and numbered lists, namely Ctrl + . and Ctrl + - . Is there something similar for PowerPoint where the work with lists is also very common? Haven't found anything here: Use keyboard shortcuts in PowerPoint. keyboard-shortcuts microsoft-powerpoint share | improve this question edited Aug 14 '14 at 14:49 Linger 2,826 10 28 40
            Read more

            How to put 3 figures in Latex with 2 figures side by side and 1 below these side by side images but in...

            Image
            1 0 I want to put 3 images in LateX such as 2 figures are side by side horizotally and 3rd figure below these 2 side by side figures but in middle. I have following code: begin{figure}[H] centering begin{minipage}{.5textwidth} centering includegraphics[width=.50linewidth]{Text/Images/Genelec_8010_AP.jpg} captionof{figure}{Genelec 8010 AP} label{fig:Genelec 8010} end{minipage}% begin{minipage}{.5textwidth} centering includegraphics[width=.69linewidth]{Text/Images/Genelec_8020_CPM.jpg} captionof{figure}{Genelec 8020 CPM} label{fig:Genelec 8020} end{minipage}% end{figure} %%Here is my 3rd picture code begin{figure} centering includegraphics[width=5cm]{Text/Images/Genelec_8030_BPM.jpg} caption{Genelec 8030 BPM} label{fig:Genelec 8030} end{figure} I want to ask ho
            Read more