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?



Extreme points of convex compact sets
A counterexample is given in the following paper:



> 
> Roberts, James W. "[A compact convex set with no extreme points](https://eudml.org/doc/218141)."
>  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.


