There are examples that show the set of extreme points of a compact convex subset of a locally convex topological vector space need not be closed when the real dimension of the space is at least 3. Is it true that the set of extreme points of a compact convex subset must be closed if the locally convex space in question has dimension 2?

By definition, a nonextreme boundary point lies on an open line segment contained in the set, which happens to be an open subset of the boundary in two dimensions. Hence the set of extreme points is a closed subset of the boundary. 


[Just a historical remark.] AFAIK, the fact that the set of all extreme points of a compact convex subset of $\mathbb{R}^{2}$ must be closed is due to the legendary American mathematician G. Baley Price (19052006), in "On the extreme points of convex sets", Duke Math. J. Volume 3, Number 1 (1937), 5667 (page 62). 

