Explaining Defitions of Minkowski functional and Gauge Functional












2












$begingroup$


I'm having trouble understanding the definition of Minkowski functional.




Let $K$ be a symmetric (i.e. if it contains $x$ it also contains $-x$) convex body in a linear space $V$. We define a function $p$ on $V$ as
$$p(x) = inf { lambda in mathbb{R}_{> 0} : x in lambda K }$$
Where $p$ is called the Minkowski functional.




Lax's gives the defintion of gauge (with respect to origin) as follows:




If $K subset V$ is a convex set in a vector space with an interior point, the gauge $p_K$ is given by:
$$p_K(x) = inf a quad a>0,frac{x}{a} in K$$




I'm having trouble breaking down the definitions in "plain english". My attempt:




  1. Minkowski functional -- choose a point in $K$. Take all reals, $lambda$. $lambda K$ is then a "scaled version of $K$". $p(x)$ is the "smallest" $lambda$ such that $x$ is still in $lambda K$. If I think about a unit ball at the origin and $x$ being the origin, is $p(x) = 0$? The set of all $lambda$ , according to my understanding, will be all positive reals - the inf of which is zero.

  2. Gauge -- I'm getting mixed up by the effect of $a$ in the denominator.






functional-analysis soft-question convex-analysis







share|cite|improve this question









$endgroup$

















    2












    $begingroup$


    I'm having trouble understanding the definition of Minkowski functional.




    Let $K$ be a symmetric (i.e. if it contains $x$ it also contains $-x$) convex body in a linear space $V$. We define a function $p$ on $V$ as
    $$p(x) = inf { lambda in mathbb{R}_{> 0} : x in lambda K }$$
    Where $p$ is called the Minkowski functional.




    Lax's gives the defintion of gauge (with respect to origin) as follows:




    If $K subset V$ is a convex set in a vector space with an interior point, the gauge $p_K$ is given by:
    $$p_K(x) = inf a quad a>0,frac{x}{a} in K$$




    I'm having trouble breaking down the definitions in "plain english". My attempt:




    1. Minkowski functional -- choose a point in $K$. Take all reals, $lambda$. $lambda K$ is then a "scaled version of $K$". $p(x)$ is the "smallest" $lambda$ such that $x$ is still in $lambda K$. If I think about a unit ball at the origin and $x$ being the origin, is $p(x) = 0$? The set of all $lambda$ , according to my understanding, will be all positive reals - the inf of which is zero.

    2. Gauge -- I'm getting mixed up by the effect of $a$ in the denominator.






    functional-analysis soft-question convex-analysis







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      I'm having trouble understanding the definition of Minkowski functional.




      Let $K$ be a symmetric (i.e. if it contains $x$ it also contains $-x$) convex body in a linear space $V$. We define a function $p$ on $V$ as
      $$p(x) = inf { lambda in mathbb{R}_{> 0} : x in lambda K }$$
      Where $p$ is called the Minkowski functional.




      Lax's gives the defintion of gauge (with respect to origin) as follows:




      If $K subset V$ is a convex set in a vector space with an interior point, the gauge $p_K$ is given by:
      $$p_K(x) = inf a quad a>0,frac{x}{a} in K$$




      I'm having trouble breaking down the definitions in "plain english". My attempt:




      1. Minkowski functional -- choose a point in $K$. Take all reals, $lambda$. $lambda K$ is then a "scaled version of $K$". $p(x)$ is the "smallest" $lambda$ such that $x$ is still in $lambda K$. If I think about a unit ball at the origin and $x$ being the origin, is $p(x) = 0$? The set of all $lambda$ , according to my understanding, will be all positive reals - the inf of which is zero.

      2. Gauge -- I'm getting mixed up by the effect of $a$ in the denominator.






      functional-analysis soft-question convex-analysis







      share|cite|improve this question









      $endgroup$




      I'm having trouble understanding the definition of Minkowski functional.




      Let $K$ be a symmetric (i.e. if it contains $x$ it also contains $-x$) convex body in a linear space $V$. We define a function $p$ on $V$ as
      $$p(x) = inf { lambda in mathbb{R}_{> 0} : x in lambda K }$$
      Where $p$ is called the Minkowski functional.




      Lax's gives the defintion of gauge (with respect to origin) as follows:




      If $K subset V$ is a convex set in a vector space with an interior point, the gauge $p_K$ is given by:
      $$p_K(x) = inf a quad a>0,frac{x}{a} in K$$




      I'm having trouble breaking down the definitions in "plain english". My attempt:




      1. Minkowski functional -- choose a point in $K$. Take all reals, $lambda$. $lambda K$ is then a "scaled version of $K$". $p(x)$ is the "smallest" $lambda$ such that $x$ is still in $lambda K$. If I think about a unit ball at the origin and $x$ being the origin, is $p(x) = 0$? The set of all $lambda$ , according to my understanding, will be all positive reals - the inf of which is zero.

      2. Gauge -- I'm getting mixed up by the effect of $a$ in the denominator.






      functional-analysis soft-question convex-analysis




      functional-analysis soft-question convex-analysis






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 22 '18 at 1:44









      yoshiyoshi

      1,242817




      1,242817






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          First of all, I think the key to the trouble of your understanding is the following relation:
          $$ forall a>0; frac{x}{a} in K iff x in aK $$
          So the definition you wrote for the Minkowski functional and the gauge Lax defines are essentially the same, note that you missed some key esssential properties: In the definition for the Minkowski functional you have the assume that $0 in K$ is an interior point (we call such sets absorbing), since only then every vector $v in V$ has some scalar small enough such that $v in lambda K$, otherwise the infimum maybe empty. The same thing goes for gauge: Lax explicitly mentions that he assumes the interior point is $0$.



          Note that your intuition in 1) is reasoable (yes $p(0)=0$), but we are not only talking about points in $K$, but in the whole of $V$, this is used in the Geometric Version of the Hahn Banach Theorem for example.






          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%2f3049058%2fexplaining-defitions-of-minkowski-functional-and-gauge-functional%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












            $begingroup$

            First of all, I think the key to the trouble of your understanding is the following relation:
            $$ forall a>0; frac{x}{a} in K iff x in aK $$
            So the definition you wrote for the Minkowski functional and the gauge Lax defines are essentially the same, note that you missed some key esssential properties: In the definition for the Minkowski functional you have the assume that $0 in K$ is an interior point (we call such sets absorbing), since only then every vector $v in V$ has some scalar small enough such that $v in lambda K$, otherwise the infimum maybe empty. The same thing goes for gauge: Lax explicitly mentions that he assumes the interior point is $0$.



            Note that your intuition in 1) is reasoable (yes $p(0)=0$), but we are not only talking about points in $K$, but in the whole of $V$, this is used in the Geometric Version of the Hahn Banach Theorem for example.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              First of all, I think the key to the trouble of your understanding is the following relation:
              $$ forall a>0; frac{x}{a} in K iff x in aK $$
              So the definition you wrote for the Minkowski functional and the gauge Lax defines are essentially the same, note that you missed some key esssential properties: In the definition for the Minkowski functional you have the assume that $0 in K$ is an interior point (we call such sets absorbing), since only then every vector $v in V$ has some scalar small enough such that $v in lambda K$, otherwise the infimum maybe empty. The same thing goes for gauge: Lax explicitly mentions that he assumes the interior point is $0$.



              Note that your intuition in 1) is reasoable (yes $p(0)=0$), but we are not only talking about points in $K$, but in the whole of $V$, this is used in the Geometric Version of the Hahn Banach Theorem for example.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                First of all, I think the key to the trouble of your understanding is the following relation:
                $$ forall a>0; frac{x}{a} in K iff x in aK $$
                So the definition you wrote for the Minkowski functional and the gauge Lax defines are essentially the same, note that you missed some key esssential properties: In the definition for the Minkowski functional you have the assume that $0 in K$ is an interior point (we call such sets absorbing), since only then every vector $v in V$ has some scalar small enough such that $v in lambda K$, otherwise the infimum maybe empty. The same thing goes for gauge: Lax explicitly mentions that he assumes the interior point is $0$.



                Note that your intuition in 1) is reasoable (yes $p(0)=0$), but we are not only talking about points in $K$, but in the whole of $V$, this is used in the Geometric Version of the Hahn Banach Theorem for example.






                share|cite|improve this answer









                $endgroup$



                First of all, I think the key to the trouble of your understanding is the following relation:
                $$ forall a>0; frac{x}{a} in K iff x in aK $$
                So the definition you wrote for the Minkowski functional and the gauge Lax defines are essentially the same, note that you missed some key esssential properties: In the definition for the Minkowski functional you have the assume that $0 in K$ is an interior point (we call such sets absorbing), since only then every vector $v in V$ has some scalar small enough such that $v in lambda K$, otherwise the infimum maybe empty. The same thing goes for gauge: Lax explicitly mentions that he assumes the interior point is $0$.



                Note that your intuition in 1) is reasoable (yes $p(0)=0$), but we are not only talking about points in $K$, but in the whole of $V$, this is used in the Geometric Version of the Hahn Banach Theorem for example.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 22 '18 at 7:30









                pitariverpitariver

                454113




                454113






























                    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%2f3049058%2fexplaining-defitions-of-minkowski-functional-and-gauge-functional%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

                    Wiesbaden

                    Image
                    Wappen Deutschlandkarte 50.0825 8.24 117 Koordinaten: 50° 5′  N , 8° 14′  O Basisdaten Bundesland: Hessen Regierungsbezirk: Darmstadt Höhe: 117 m ü. NHN Fläche: 203,93 km 2 Einwohner: 278.654 (31. Dez. 2017) [1] Bevölkerungsdichte: 1366 Einwohner je km 2 Postleitzahlen: 65183–65207, 55246, 55252 Vorlage:Infobox Gemeinde in Deutschland/Wartung/PLZ enthält Text Vorwahlen: 0611, 06122, 06127, 06134 Kfz-Kennzeichen: WI Gemeindeschlüssel: 06 4 14 000 LOCODE: DE WIB NUTS: DE714 Stadtgliederung: 26 Ortsbezirke Adresse der Stadtverwaltung: Schlossplatz 6 65183 Wiesbaden Webpräsenz: www.wiesbaden.de Oberbürgermeister: Sven Gerich (SPD) Lage der Landeshauptstadt Wiesbaden in Hessen und im Regierungsbezirk Darmstadt Offizielles Logo der Landeshauptstadt Wiesbaden Flagge der Landeshauptstadt Wiesbaden Wiesbaden is
                    Read more

                    Marschland

                    Image
                    Kilometerweite Marsch in den Vier- und Marschlanden am Elbdeich Die Wedeler Marschlandschaft in Schleswig-Holstein Der Hadelner Kanal zur Entwässerung der Marsch Die tief gelegene Marsch und ein Entwässerungskanal Das Schöpfwerk Otterndorf im Land Hadeln zur Entwässerung der Medem verfügte damals bereits über die größte Kreiselpumpe Europas. Das Schöpfwerk in Neuhaus für die Aue und des Neuhaus-Bülkauer Kanals Als Marsch(land) (v. niederdt., altsächs. mersc ) – auch Masch , Mersch oder Schwemmland genannt – bezeichnet man eine nacheiszeitlich entstandene geomorphologische Landform im Gebiet der nordwestdeutschen Küsten und Flüsse sowie vergleichbare Landformen weltweit. Inhaltsverzeichnis 1 Landschaft 2 Entstehung 3 Marschböden 4 Nutzung 5 Marschgebiete in Deutschland 6 Siehe auch 7 Literatur 8 Weblinks 9 Einzelnachweise Landschaft | Marschen sind generell flache Landstriche ohne natürliche Erhebungen. Si
                    Read more

                    Dieringhausen

                    Image
                    Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
                    Read more