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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$\begingroup$ Retagged the question – banach-spaces out, convexity in. $\endgroup$– Harald Hanche-OlsenCommented Jan 5, 2010 at 14:11
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$\begingroup$ 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. $\endgroup$– Pete L. ClarkCommented Jan 5, 2010 at 16:55
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1$\begingroup$ @Pete For instance, the convex hull of the four points $(\pm 1, \pm 1, 0)$ and the unit circle in the $x$-$z$ plane. $\endgroup$– Greg KuperbergCommented Apr 3, 2011 at 20:43
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$\begingroup$ 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. $\endgroup$– petrbelCommented Nov 3, 2014 at 19:02
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2 Answers
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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).
