Let $X$ be a probability measure space and $\psi_1....\psi_N$ measurable, real valued
functions on $X$. Consider all possible partitions of $X$ into $N$ disjoint subdomains $X_i$
on which $\int_{X_i}\psi_i$ are prescribed. I'll address the question of characterizing the set $(m_1,,,m_N) \in \mathbb{R}^N$ for which there exists a partition $X_1, \ldots X_N$ of $X$
satisfying $\int_{X_i}\psi_i= m_i$ and discuss some optimization problems on this set of partitions.
The relation of this problem to semi-discrete version of optimal mass transportation is discussed, as well as applications to game theory.
