On Semi-discrete Monge–Kantorovich and Generalized Partitions

We address the question of characterizing the set of points in $$m$$m dimensional space for which there exists a partition of a given measure space into $$m$$m essentially disjoint sets satisfying prescribed integral conditions. In addition, we discuss some optimization problems on this set of partitions. The relation of this problem to the semi-discrete version of optimal mass transportation is discussed as well.