Explaining Defitions of Minkowski functional and Gauge Functional












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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











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      asked Dec 22 '18 at 1:44









      yoshiyoshi

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          $begingroup$

          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









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            $begingroup$

            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









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              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

                444213




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