Extreme points of a compact convex set are a $G_\delta$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:16:49Z http://mathoverflow.net/feeds/question/94715 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94715/extreme-points-of-a-compact-convex-set-are-a-g-delta Extreme points of a compact convex set are a $G_\delta$? Anthony Quas 2012-04-21T07:13:49Z 2012-04-21T08:26:46Z <p>Dear All,</p> <p>I'm reading a paper (<em>Residuality of Dynamical Morphisms</em> 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. </p> <p>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 <i>a general theorem states that the extreme points of a compact convex set form a $G_\delta$</i>. They don't say whose general theorem! I've looked reasonably hard for a suitable reference without success. Can anyone give me any pointers?</p> <p>Thanks...</p> http://mathoverflow.net/questions/94715/extreme-points-of-a-compact-convex-set-are-a-g-delta/94718#94718 Answer by Pietro Majer for Extreme points of a compact convex set are a $G_\delta$? Pietro Majer 2012-04-21T07:44:15Z 2012-04-21T08:26:46Z <p>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, <em>The representation of linear functionals by measures on sets of extreme points</em>, Ann.Inst. Fourier (Grenoble) (1959) . A very good reference for these topics is Phelp's LNM <em>Lectures on the Choquet's theorem</em> (2001).</p>