On semi-discrete Monge problem and generalized optimal partitions

TL;DR

This paper investigates the semi-discrete Monge problem, characterizing partitions of a measure space with prescribed integrals, and explores related optimization problems and connections to semi-discrete optimal mass transportation.

Contribution

It provides a characterization of feasible integral vectors for partitions and discusses optimization and connections to semi-discrete optimal transport.

Findings

Characterization of feasible integral vectors for partitions.

Analysis of optimization problems on partition sets.

Discussion of relations to semi-discrete optimal mass transportation.

Abstract

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.