Closed convex hull of a subset of $mathbb{C}^d$












1












$begingroup$


Let $W$ be a subset of $mathbb{C}^d$ and $co (overline{W})$ be the closed convex hull of $W$ (here $overline{W}$ is the closure of $W$ with respect to the topology of $mathbb{C}^d$).




I don't understand what we mean by the closed convex hull of $W$? Also is it true that
$$co (overline{W}) subseteq W ?$$




My goal is to compare



$$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in W}$$
and
$$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in co (overline{W})},$$
where
$$|lambda|_2:=left(displaystylesum_{k=1}^d|lambda_k|^2right)^{1/2}.$$






general-topology convex-hulls







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    Let $W$ be a subset of $mathbb{C}^d$ and $co (overline{W})$ be the closed convex hull of $W$ (here $overline{W}$ is the closure of $W$ with respect to the topology of $mathbb{C}^d$).




    I don't understand what we mean by the closed convex hull of $W$? Also is it true that
    $$co (overline{W}) subseteq W ?$$




    My goal is to compare



    $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in W}$$
    and
    $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in co (overline{W})},$$
    where
    $$|lambda|_2:=left(displaystylesum_{k=1}^d|lambda_k|^2right)^{1/2}.$$






    general-topology convex-hulls







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      Let $W$ be a subset of $mathbb{C}^d$ and $co (overline{W})$ be the closed convex hull of $W$ (here $overline{W}$ is the closure of $W$ with respect to the topology of $mathbb{C}^d$).




      I don't understand what we mean by the closed convex hull of $W$? Also is it true that
      $$co (overline{W}) subseteq W ?$$




      My goal is to compare



      $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in W}$$
      and
      $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in co (overline{W})},$$
      where
      $$|lambda|_2:=left(displaystylesum_{k=1}^d|lambda_k|^2right)^{1/2}.$$






      general-topology convex-hulls







      share|cite|improve this question











      $endgroup$




      Let $W$ be a subset of $mathbb{C}^d$ and $co (overline{W})$ be the closed convex hull of $W$ (here $overline{W}$ is the closure of $W$ with respect to the topology of $mathbb{C}^d$).




      I don't understand what we mean by the closed convex hull of $W$? Also is it true that
      $$co (overline{W}) subseteq W ?$$




      My goal is to compare



      $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in W}$$
      and
      $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in co (overline{W})},$$
      where
      $$|lambda|_2:=left(displaystylesum_{k=1}^d|lambda_k|^2right)^{1/2}.$$






      general-topology convex-hulls




      general-topology convex-hulls






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 27 '18 at 19:23







      Schüler

















      asked Dec 27 '18 at 19:15









      SchülerSchüler

      1,5391421




      1,5391421






















          1 Answer
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          $begingroup$

          The closed convex hull of a set $W$ is the smallest closed convex set containing $W$. Let $R=sup{|lambda|:lambdain W}$. Then the ball around $0$ of radius $R$ is closed and convex and contains $W$. Hence $mbox{co}(overline{W})subset B_R(0)$, so $sup{|lambda|:lambdainmbox{co}(overline{W})}leq R$. Since also $Wsubsetmbox{co}(overline{W})$ we find $sup{|lambda|:lambdainmbox{co}(overline{W})}=R$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Please what is the difference between closed convex hull and convex hull? Thanks a lot.
            $endgroup$
            – Schüler
            Dec 28 '18 at 5:07










          • $begingroup$
            Closed convex hull is the smallest closed convex set containing $W$. Convex hull is smallest convex set containing $W$.
            $endgroup$
            – SmileyCraft
            Dec 28 '18 at 13:01










          • $begingroup$
            Please it is true that $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in W}=sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in overline{W}}? $$ Thanks a lot.
            $endgroup$
            – Schüler
            Dec 28 '18 at 13:06










          • $begingroup$
            That is true indeed. The closure $overline{W}$ is the smallest closed set containing $W$. The argument for equality of supremum is actually the same as for $mbox{co}(overline{W})$. Try it! :)
            $endgroup$
            – SmileyCraft
            Dec 28 '18 at 14:18











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          1 Answer
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          oldest

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          1 Answer
          1






          active

          oldest

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          active

          oldest

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          active

          oldest

          votes









          2












          $begingroup$

          The closed convex hull of a set $W$ is the smallest closed convex set containing $W$. Let $R=sup{|lambda|:lambdain W}$. Then the ball around $0$ of radius $R$ is closed and convex and contains $W$. Hence $mbox{co}(overline{W})subset B_R(0)$, so $sup{|lambda|:lambdainmbox{co}(overline{W})}leq R$. Since also $Wsubsetmbox{co}(overline{W})$ we find $sup{|lambda|:lambdainmbox{co}(overline{W})}=R$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Please what is the difference between closed convex hull and convex hull? Thanks a lot.
            $endgroup$
            – Schüler
            Dec 28 '18 at 5:07










          • $begingroup$
            Closed convex hull is the smallest closed convex set containing $W$. Convex hull is smallest convex set containing $W$.
            $endgroup$
            – SmileyCraft
            Dec 28 '18 at 13:01










          • $begingroup$
            Please it is true that $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in W}=sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in overline{W}}? $$ Thanks a lot.
            $endgroup$
            – Schüler
            Dec 28 '18 at 13:06










          • $begingroup$
            That is true indeed. The closure $overline{W}$ is the smallest closed set containing $W$. The argument for equality of supremum is actually the same as for $mbox{co}(overline{W})$. Try it! :)
            $endgroup$
            – SmileyCraft
            Dec 28 '18 at 14:18
















          2












          $begingroup$

          The closed convex hull of a set $W$ is the smallest closed convex set containing $W$. Let $R=sup{|lambda|:lambdain W}$. Then the ball around $0$ of radius $R$ is closed and convex and contains $W$. Hence $mbox{co}(overline{W})subset B_R(0)$, so $sup{|lambda|:lambdainmbox{co}(overline{W})}leq R$. Since also $Wsubsetmbox{co}(overline{W})$ we find $sup{|lambda|:lambdainmbox{co}(overline{W})}=R$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Please what is the difference between closed convex hull and convex hull? Thanks a lot.
            $endgroup$
            – Schüler
            Dec 28 '18 at 5:07










          • $begingroup$
            Closed convex hull is the smallest closed convex set containing $W$. Convex hull is smallest convex set containing $W$.
            $endgroup$
            – SmileyCraft
            Dec 28 '18 at 13:01










          • $begingroup$
            Please it is true that $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in W}=sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in overline{W}}? $$ Thanks a lot.
            $endgroup$
            – Schüler
            Dec 28 '18 at 13:06










          • $begingroup$
            That is true indeed. The closure $overline{W}$ is the smallest closed set containing $W$. The argument for equality of supremum is actually the same as for $mbox{co}(overline{W})$. Try it! :)
            $endgroup$
            – SmileyCraft
            Dec 28 '18 at 14:18














          2












          2








          2





          $begingroup$

          The closed convex hull of a set $W$ is the smallest closed convex set containing $W$. Let $R=sup{|lambda|:lambdain W}$. Then the ball around $0$ of radius $R$ is closed and convex and contains $W$. Hence $mbox{co}(overline{W})subset B_R(0)$, so $sup{|lambda|:lambdainmbox{co}(overline{W})}leq R$. Since also $Wsubsetmbox{co}(overline{W})$ we find $sup{|lambda|:lambdainmbox{co}(overline{W})}=R$.






          share|cite|improve this answer









          $endgroup$



          The closed convex hull of a set $W$ is the smallest closed convex set containing $W$. Let $R=sup{|lambda|:lambdain W}$. Then the ball around $0$ of radius $R$ is closed and convex and contains $W$. Hence $mbox{co}(overline{W})subset B_R(0)$, so $sup{|lambda|:lambdainmbox{co}(overline{W})}leq R$. Since also $Wsubsetmbox{co}(overline{W})$ we find $sup{|lambda|:lambdainmbox{co}(overline{W})}=R$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 27 '18 at 19:54









          SmileyCraftSmileyCraft

          3,739519




          3,739519












          • $begingroup$
            Please what is the difference between closed convex hull and convex hull? Thanks a lot.
            $endgroup$
            – Schüler
            Dec 28 '18 at 5:07










          • $begingroup$
            Closed convex hull is the smallest closed convex set containing $W$. Convex hull is smallest convex set containing $W$.
            $endgroup$
            – SmileyCraft
            Dec 28 '18 at 13:01










          • $begingroup$
            Please it is true that $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in W}=sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in overline{W}}? $$ Thanks a lot.
            $endgroup$
            – Schüler
            Dec 28 '18 at 13:06










          • $begingroup$
            That is true indeed. The closure $overline{W}$ is the smallest closed set containing $W$. The argument for equality of supremum is actually the same as for $mbox{co}(overline{W})$. Try it! :)
            $endgroup$
            – SmileyCraft
            Dec 28 '18 at 14:18


















          • $begingroup$
            Please what is the difference between closed convex hull and convex hull? Thanks a lot.
            $endgroup$
            – Schüler
            Dec 28 '18 at 5:07










          • $begingroup$
            Closed convex hull is the smallest closed convex set containing $W$. Convex hull is smallest convex set containing $W$.
            $endgroup$
            – SmileyCraft
            Dec 28 '18 at 13:01










          • $begingroup$
            Please it is true that $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in W}=sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in overline{W}}? $$ Thanks a lot.
            $endgroup$
            – Schüler
            Dec 28 '18 at 13:06










          • $begingroup$
            That is true indeed. The closure $overline{W}$ is the smallest closed set containing $W$. The argument for equality of supremum is actually the same as for $mbox{co}(overline{W})$. Try it! :)
            $endgroup$
            – SmileyCraft
            Dec 28 '18 at 14:18
















          $begingroup$
          Please what is the difference between closed convex hull and convex hull? Thanks a lot.
          $endgroup$
          – Schüler
          Dec 28 '18 at 5:07




          $begingroup$
          Please what is the difference between closed convex hull and convex hull? Thanks a lot.
          $endgroup$
          – Schüler
          Dec 28 '18 at 5:07












          $begingroup$
          Closed convex hull is the smallest closed convex set containing $W$. Convex hull is smallest convex set containing $W$.
          $endgroup$
          – SmileyCraft
          Dec 28 '18 at 13:01




          $begingroup$
          Closed convex hull is the smallest closed convex set containing $W$. Convex hull is smallest convex set containing $W$.
          $endgroup$
          – SmileyCraft
          Dec 28 '18 at 13:01












          $begingroup$
          Please it is true that $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in W}=sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in overline{W}}? $$ Thanks a lot.
          $endgroup$
          – Schüler
          Dec 28 '18 at 13:06




          $begingroup$
          Please it is true that $$sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in W}=sup{|lambda|_2,;lambda=(lambda_1,cdots,lambda_d) in overline{W}}? $$ Thanks a lot.
          $endgroup$
          – Schüler
          Dec 28 '18 at 13:06












          $begingroup$
          That is true indeed. The closure $overline{W}$ is the smallest closed set containing $W$. The argument for equality of supremum is actually the same as for $mbox{co}(overline{W})$. Try it! :)
          $endgroup$
          – SmileyCraft
          Dec 28 '18 at 14:18




          $begingroup$
          That is true indeed. The closure $overline{W}$ is the smallest closed set containing $W$. The argument for equality of supremum is actually the same as for $mbox{co}(overline{W})$. Try it! :)
          $endgroup$
          – SmileyCraft
          Dec 28 '18 at 14:18


















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