# Uniqueness of optimal solutions for semi-discrete transport with p-norm cost functions

## TL;DR

This paper investigates conditions under which semi-discrete optimal transport solutions can be characterized by shifts that partition the continuous region, providing examples of failures and identifying classes where partitioning always occurs.

## Contribution

It identifies when shift-characterized solutions in semi-discrete transport always form partitions, expanding understanding of solution structures in optimal transport problems.

## Key findings

- Examples where partitioning fails in semi-discrete transport.
- A large class of problems where shift solutions always partition.
- Insights into the structure of optimal solutions in semi-discrete transport.

## Abstract

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.