On semi-discrete Monge problem and generalized optimal partitions

Let $X$ a probability measure space and $ψ_1....ψ_N$ measurable, real valued functions on $X$. Consider all possible partitions of $X$ into $N$ disjoint subdomains $X_i$ on which $\int_{X_i}ψ_i$ 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 $\int_{X_i}ψ_i= m_i$ 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.

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