Extreme points of convex compact sets

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Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an extreme point. Is it still unknown?






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    Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an extreme point. Is it still unknown?






    reference-request fa.functional-analysis convexity extreme-points krein-milman-theorem







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      up vote
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      up vote
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      down vote

      favorite











      Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an extreme point. Is it still unknown?






      reference-request fa.functional-analysis convexity extreme-points krein-milman-theorem







      share|cite|improve this question













      Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an extreme point. Is it still unknown?






      reference-request fa.functional-analysis convexity extreme-points krein-milman-theorem




      reference-request fa.functional-analysis convexity extreme-points krein-milman-theorem






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      asked 59 mins ago









      Fedor Petrov

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          A counterexample is given in the following paper:




          Roberts, James W. "A compact convex set with no extreme points."
          Studia Mathematica 60.3 (1977): 255-266.




          I did not see the original paper, but an exposition can be found in Section 5.6 of the book "Metric linear spaces" by Rolewicz.






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            eudml.org/doc/218141 gives me a link to pdf
            – Dima Pasechnik
            23 mins ago










          • @Dima Thank you; I added the link.
            – Michael Greinecker
            21 mins ago










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          1 Answer
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          1 Answer
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          up vote
          6
          down vote













          A counterexample is given in the following paper:




          Roberts, James W. "A compact convex set with no extreme points."
          Studia Mathematica 60.3 (1977): 255-266.




          I did not see the original paper, but an exposition can be found in Section 5.6 of the book "Metric linear spaces" by Rolewicz.






          share|cite|improve this answer


















          • 2




            eudml.org/doc/218141 gives me a link to pdf
            – Dima Pasechnik
            23 mins ago










          • @Dima Thank you; I added the link.
            – Michael Greinecker
            21 mins ago














          up vote
          6
          down vote













          A counterexample is given in the following paper:




          Roberts, James W. "A compact convex set with no extreme points."
          Studia Mathematica 60.3 (1977): 255-266.




          I did not see the original paper, but an exposition can be found in Section 5.6 of the book "Metric linear spaces" by Rolewicz.






          share|cite|improve this answer


















          • 2




            eudml.org/doc/218141 gives me a link to pdf
            – Dima Pasechnik
            23 mins ago










          • @Dima Thank you; I added the link.
            – Michael Greinecker
            21 mins ago












          up vote
          6
          down vote










          up vote
          6
          down vote









          A counterexample is given in the following paper:




          Roberts, James W. "A compact convex set with no extreme points."
          Studia Mathematica 60.3 (1977): 255-266.




          I did not see the original paper, but an exposition can be found in Section 5.6 of the book "Metric linear spaces" by Rolewicz.






          share|cite|improve this answer














          A counterexample is given in the following paper:




          Roberts, James W. "A compact convex set with no extreme points."
          Studia Mathematica 60.3 (1977): 255-266.




          I did not see the original paper, but an exposition can be found in Section 5.6 of the book "Metric linear spaces" by Rolewicz.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 22 mins ago

























          answered 42 mins ago









          Michael Greinecker

          7,18423554




          7,18423554







          • 2




            eudml.org/doc/218141 gives me a link to pdf
            – Dima Pasechnik
            23 mins ago










          • @Dima Thank you; I added the link.
            – Michael Greinecker
            21 mins ago












          • 2




            eudml.org/doc/218141 gives me a link to pdf
            – Dima Pasechnik
            23 mins ago










          • @Dima Thank you; I added the link.
            – Michael Greinecker
            21 mins ago







          2




          2




          eudml.org/doc/218141 gives me a link to pdf
          – Dima Pasechnik
          23 mins ago




          eudml.org/doc/218141 gives me a link to pdf
          – Dima Pasechnik
          23 mins ago












          @Dima Thank you; I added the link.
          – Michael Greinecker
          21 mins ago




          @Dima Thank you; I added the link.
          – Michael Greinecker
          21 mins ago

















           

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