# Compact Convex sets and Extreme Points

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
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 that so? Where can one find some example of that set? For me it seems impossible to exists. – petrbel Nov 3 '14 at 19:02

[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).