Show that support function of a set $S$ and support function of the closure of that set $bar{S}$ are equal.












1














Let $Ssubseteq mathbb{R}^n$.



The support function of set $S$ is defined as the following



$$
sigma_S(x)=sup_{y in S} x^Ty
$$

where $x in mathbb{R}^n$.



Show that $sigma_S(x)=sigma_{bar{S}}(x) ,,,,forall ,,,x in mathbb{R}^n$.



If the set $S$ were closed then $S=bar{S}$ so we are done. For the case that $S neqbar{S}$ we need to prove the statement. My approach is to say that the closure of a set is the set of all supporting hyperplains (which I do not know how to prove!), then show that $sigma_S(x)$ takes its value for every $x$ when $y$ is at the boundary of $S$.










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    1














    Let $Ssubseteq mathbb{R}^n$.



    The support function of set $S$ is defined as the following



    $$
    sigma_S(x)=sup_{y in S} x^Ty
    $$

    where $x in mathbb{R}^n$.



    Show that $sigma_S(x)=sigma_{bar{S}}(x) ,,,,forall ,,,x in mathbb{R}^n$.



    If the set $S$ were closed then $S=bar{S}$ so we are done. For the case that $S neqbar{S}$ we need to prove the statement. My approach is to say that the closure of a set is the set of all supporting hyperplains (which I do not know how to prove!), then show that $sigma_S(x)$ takes its value for every $x$ when $y$ is at the boundary of $S$.










    share|cite|improve this question

























      1












      1








      1







      Let $Ssubseteq mathbb{R}^n$.



      The support function of set $S$ is defined as the following



      $$
      sigma_S(x)=sup_{y in S} x^Ty
      $$

      where $x in mathbb{R}^n$.



      Show that $sigma_S(x)=sigma_{bar{S}}(x) ,,,,forall ,,,x in mathbb{R}^n$.



      If the set $S$ were closed then $S=bar{S}$ so we are done. For the case that $S neqbar{S}$ we need to prove the statement. My approach is to say that the closure of a set is the set of all supporting hyperplains (which I do not know how to prove!), then show that $sigma_S(x)$ takes its value for every $x$ when $y$ is at the boundary of $S$.










      share|cite|improve this question













      Let $Ssubseteq mathbb{R}^n$.



      The support function of set $S$ is defined as the following



      $$
      sigma_S(x)=sup_{y in S} x^Ty
      $$

      where $x in mathbb{R}^n$.



      Show that $sigma_S(x)=sigma_{bar{S}}(x) ,,,,forall ,,,x in mathbb{R}^n$.



      If the set $S$ were closed then $S=bar{S}$ so we are done. For the case that $S neqbar{S}$ we need to prove the statement. My approach is to say that the closure of a set is the set of all supporting hyperplains (which I do not know how to prove!), then show that $sigma_S(x)$ takes its value for every $x$ when $y$ is at the boundary of $S$.







      convex-analysis convex-optimization supremum-and-infimum convex-geometry convex-hulls






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      asked Nov 19 '18 at 4:56









      Sepide

      2918




      2918






















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          It is obvious that $sigma_S(x) leq sigma_{ overline {S}} (x)$. For the other inequality take $y in overline {S}$ and choose a sequence ${y_n} subset S$ such that $y_n to y$. Since $x^{T}y_n leq sigma_S(x)$ for all $n$ and $x^{T}y_n to x^{T}y$ we get $x^{T}y leq sigma_S(x)$. Taking supremum over $y$ we get $sigma_{ overline {S}} (x) leq sigma_S(x)$.






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          • I need to show the first sentence. I understand that intuitively makes sense but how we can show it?
            – Sepide
            Nov 20 '18 at 3:50










          • If $A subset B$ then $sup A leq sup B$ by definition of supremum.
            – Kavi Rama Murthy
            Nov 20 '18 at 5:32











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          It is obvious that $sigma_S(x) leq sigma_{ overline {S}} (x)$. For the other inequality take $y in overline {S}$ and choose a sequence ${y_n} subset S$ such that $y_n to y$. Since $x^{T}y_n leq sigma_S(x)$ for all $n$ and $x^{T}y_n to x^{T}y$ we get $x^{T}y leq sigma_S(x)$. Taking supremum over $y$ we get $sigma_{ overline {S}} (x) leq sigma_S(x)$.






          share|cite|improve this answer





















          • I need to show the first sentence. I understand that intuitively makes sense but how we can show it?
            – Sepide
            Nov 20 '18 at 3:50










          • If $A subset B$ then $sup A leq sup B$ by definition of supremum.
            – Kavi Rama Murthy
            Nov 20 '18 at 5:32
















          1














          It is obvious that $sigma_S(x) leq sigma_{ overline {S}} (x)$. For the other inequality take $y in overline {S}$ and choose a sequence ${y_n} subset S$ such that $y_n to y$. Since $x^{T}y_n leq sigma_S(x)$ for all $n$ and $x^{T}y_n to x^{T}y$ we get $x^{T}y leq sigma_S(x)$. Taking supremum over $y$ we get $sigma_{ overline {S}} (x) leq sigma_S(x)$.






          share|cite|improve this answer





















          • I need to show the first sentence. I understand that intuitively makes sense but how we can show it?
            – Sepide
            Nov 20 '18 at 3:50










          • If $A subset B$ then $sup A leq sup B$ by definition of supremum.
            – Kavi Rama Murthy
            Nov 20 '18 at 5:32














          1












          1








          1






          It is obvious that $sigma_S(x) leq sigma_{ overline {S}} (x)$. For the other inequality take $y in overline {S}$ and choose a sequence ${y_n} subset S$ such that $y_n to y$. Since $x^{T}y_n leq sigma_S(x)$ for all $n$ and $x^{T}y_n to x^{T}y$ we get $x^{T}y leq sigma_S(x)$. Taking supremum over $y$ we get $sigma_{ overline {S}} (x) leq sigma_S(x)$.






          share|cite|improve this answer












          It is obvious that $sigma_S(x) leq sigma_{ overline {S}} (x)$. For the other inequality take $y in overline {S}$ and choose a sequence ${y_n} subset S$ such that $y_n to y$. Since $x^{T}y_n leq sigma_S(x)$ for all $n$ and $x^{T}y_n to x^{T}y$ we get $x^{T}y leq sigma_S(x)$. Taking supremum over $y$ we get $sigma_{ overline {S}} (x) leq sigma_S(x)$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 19 '18 at 5:20









          Kavi Rama Murthy

          50.3k31854




          50.3k31854












          • I need to show the first sentence. I understand that intuitively makes sense but how we can show it?
            – Sepide
            Nov 20 '18 at 3:50










          • If $A subset B$ then $sup A leq sup B$ by definition of supremum.
            – Kavi Rama Murthy
            Nov 20 '18 at 5:32


















          • I need to show the first sentence. I understand that intuitively makes sense but how we can show it?
            – Sepide
            Nov 20 '18 at 3:50










          • If $A subset B$ then $sup A leq sup B$ by definition of supremum.
            – Kavi Rama Murthy
            Nov 20 '18 at 5:32
















          I need to show the first sentence. I understand that intuitively makes sense but how we can show it?
          – Sepide
          Nov 20 '18 at 3:50




          I need to show the first sentence. I understand that intuitively makes sense but how we can show it?
          – Sepide
          Nov 20 '18 at 3:50












          If $A subset B$ then $sup A leq sup B$ by definition of supremum.
          – Kavi Rama Murthy
          Nov 20 '18 at 5:32




          If $A subset B$ then $sup A leq sup B$ by definition of supremum.
          – Kavi Rama Murthy
          Nov 20 '18 at 5:32


















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