## Extreme points of a compact convex set are a $G_\delta$?

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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...

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## 1 Answer

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).

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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 2012 at 8:27

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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 2012 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 2012 at 16:34
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