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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?

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Retagged the question – banach-spaces out, convexity in. –  Harald Hanche-Olsen Jan 5 '10 at 14:11
    
Thanks for your responses. I deleted my original question, which was rather silly: somehow, in your perfectly clear 2.5 line answer, I missed the part where you used the 2-dimensionality. But I think it's good to have an example of the failure of closedness in higher dimensions. –  Pete L. Clark Jan 5 '10 at 16:55
    
@Pete For instance, the convex hull of the four points $(\pm 1, \pm 1, 0)$ and the unit circle in the $x$-$z$ plane. –  Greg Kuperberg Apr 3 '11 at 20:43

2 Answers 2

up vote 8 down vote accepted

By definition, a non-extreme 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.

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[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 (1905-2006), in "On the extreme points of convex sets", Duke Math. J. Volume 3, Number 1 (1937), 56-67 (page 62).

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