Uniqueness of optimal solutions for semi-discrete transport with p-norm cost functions
For researchers in optimal transport, this clarifies when the commonly assumed partition property holds, addressing a theoretical gap.
The paper identifies conditions under which the shift-characterized solution to semi-discrete optimal transport partitions the continuous region, providing a large class of problems where partitioning always occurs, and giving counterexamples where it fails.
Semi-discrete transport can be characterized in terms of real-valued shifts. Often, but not always, the solution to the shift-characterized problem partitions the continuous region. This paper gives examples of when partitioning fails, and offers a large class of semi-discrete transport problems where the shift-characterized solution is always a partition.