Let $\Omega$ be a convex compact set in $\mathbb{R}^n$, $f\colon \Omega \to \mathbb{R}$ be a convex function. Consider an optimization problem $ \int\limits_{\Omega}f(x)\,\mu(dx) \to \max\limits_{\mu \in M}, $ where $M$ is a set of Borel probability measures on $\Omega$ such that $ \int\limits_{\Omega} x \, \mu (dx) = x^{\ast} \in \mathop{\mathrm{int}}\Omega \;\;\; \text{($x^{\ast}$ is fixed)} $ How to show that if the maximization problem is solvable then there exists an optimal measure $\mu^{\ast}$ such that $ \mu^{\ast} = p_{1}\delta(x-x_{1})+\ldots+p_{n+1}\delta(x-x_{n+1}), $ where $p_{i} \geqslant 0$, $\sum_{i=1}^{n+1}p_{i}=1$ and $x_{1},\ldots,x_{n+1}$ are extreme points of $\Omega$? I tried to combine Krein-Milman and Caratheodory theorems, but without success.
Find optimal measure
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measure-theory
optimization
convex-analysis
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0@PavelM Re Re 2: yes, I think we may assume that $f$ is continuous – 2012-12-28