Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Dear All,

I'm reading a paper (Residuality of Dynamical Morphisms by Burton, Keane and Serafin) that makes a claim that I've been unable to verify or find a reference for. The claim is made that the extreme points of a compact convex set in a locally convex topological vector space form a $G_\delta$ subset of the space.

I've been able to verify it in the specific context of the paper (sets of invariant measures for a continuous transformation of a compact metric space), but in the article they say a general theorem states that the extreme points of a compact convex set form a $G_\delta$. They don't say whose general theorem! I've looked reasonably hard for a suitable reference without success. Can anyone give me any pointers?

Thanks...

share|improve this question
add comment

1 Answer

up vote 7 down vote accepted

For a non-metrizable compact convex subset of a locally convex space, extreme points need not even form a Borel set. This has been shown by Bishop-de Leeuw, The representation of linear functionals by measures on sets of extreme points, Ann.Inst. Fourier (Grenoble) (1959) . A very good reference for these topics is Phelp's LNM Lectures on the Choquet's theorem (2001).

share|improve this answer
3  
The metrizable case is quite straightforward: Fix a compatible metric $d$ on $K$. Let $$F_n = \left\{x \in K\,:\, \text{there are } y,z \in K\text{ such that }x = \frac{1}{2}(y+z)\text{ and }d(y,z) \geq \frac{1}{n}\right\}.$$ Then $F_n$ is closed and a point is non-extremal if and only if it is in $F = \bigcup_n F_n$. Thus the set of extremal points $\operatorname{ex}{K} = K \smallsetminus F$ is a $G_\delta$. –  Theo Buehler Apr 21 '12 at 8:27
1  
Here's a link to the paper by Bishop-de Leeuw: numdam.org/item?id=AIF_1959__9__305_0 The counterexample appears in section VII. –  Theo Buehler Apr 21 '12 at 8:39
    
Thanks Theo and Pietro. I actually tried to write down some open sets along these lines that intersected to the extreme points without success, but anyway this is clear now. –  Anthony Quas Apr 21 '12 at 16:34
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.