Extreme points of a set of probability measures - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T04:23:39Z http://mathoverflow.net/feeds/question/85527 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85527/extreme-points-of-a-set-of-probability-measures Extreme points of a set of probability measures Henrique de Oliveira 2012-01-12T21:24:30Z 2012-01-12T22:37:16Z <p>Consider the set of Borel-measurable probability measures over the interval $[0,1]$ with a given mean, say 1/2. To be precise, I'm talking about the following set $$M=\left(\mu\in \Delta([0,1]):\int x\mu(dx)=1/2\right)$$</p> <p>This is a convex set, with convex combinations defined as $\mu=\alpha \eta +(1-\alpha)\xi$ when $\mu(E)=\alpha \eta(E)+(1-\alpha)\xi(E)$ for every Borel set $E\subset[0,1]$. </p> <p>My question is: what is the set of extreme points of this convex set?</p> <p>My conjecture is that it is the subset of probability measures with support consisting of at most two points. I'm fairly convinced that this is true, but so far I haven't been able to come up with an elegant argument to show this and I thought that perhaps this is a well-known result. It is not hard to show for the case of atomless measures or for the case of discrete measures, but I'd like to have the general case.</p> http://mathoverflow.net/questions/85527/extreme-points-of-a-set-of-probability-measures/85530#85530 Answer by Igor Rivin for Extreme points of a set of probability measures Igor Rivin 2012-01-12T22:01:30Z 2012-01-12T22:37:16Z <p>This question (where you prescribe a set of moments, on an arbitrary measure space) is completely answered in <a href="http://dl.dropbox.com/u/5188175/971911.pdf" rel="nofollow">this very cool paper.</a> (G. Winkler, Extremal points of moment sets).</p>