Preparing to a lecture on KreinMilman 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?
1 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): 255266.
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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