Timeline for Maximizing linear function (not necessarily continuous) over a compact, closed and convex domain

4 events

when toggle format	what		by	license	comment
Aug 4, 2017 at 2:30	vote	accept	Boby
Aug 4, 2017 at 2:13	comment	added	Joel Moreira		I was thinking of the version of Choquet's theorem which says that every point of a compact convex set ${\mathcal D}$ is the barycenter of a probability measure on the extreme points of ${\mathcal D}$, although I realize now that is not exactly the version on wikipedia. But the version on wikipedia is good enough to answer your question: Let $F:{\mathcal D}\to{\mathbb R}$ be the affine function $F:\mu\mapsto\int_{\mathbb R}f(x)d\mu(x)$. Then we have $F(\mu)=\int_{\mathcal E}F(\rho)d\nu(\rho)\leq\sup_{\rho\in{\mathcal E}}F(\rho)$.
Aug 4, 2017 at 1:57	comment	added	Boby		The Choquet's theorem states if $D$ is compact and convex and if $f$ is an affine function on $D$, then there exists a probability measure $\nu$ supported on $\mathcal{E}$ such that $f(c)=\int_{\mathcal{E}} f(e) d\rho(e)$. I have a question. How are you getting $\mu = \int_{\mathcal{E}} e d\nu(e)$. I don't think $\mu$ is an affince function. Can you explain this point a bit.
Aug 3, 2017 at 18:48	history	answered	Joel Moreira	CC BY-SA 3.0