  Let $X$ 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. We
address the question of characterizing the set $(m_1,,,m_N) \in \R^N$ for which
there exists a partition $X_1, ... X_N$ 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.
