# Geometry of Kantorovich Polytopes and Support of Optimizers for Repulsive Multi-Marginal Optimal Transport on Finite State Spaces

## TL;DR

This paper explores the geometric structure of Kantorovich polytopes in symmetric multi-marginal optimal transport problems on finite spaces, providing computational insights, theoretical support conditions, and connections to Monge solutions.

## Contribution

It offers a detailed analysis of Kantorovich polytopes, computes all extreme points for small cases, and establishes a support condition for optimal solutions that links Kantorovich and Monge formulations.

## Key findings

- All extreme points of the Kantorovich polytope can be computed for small N and .
- A necessary support condition for optimal solutions is derived, extending beyond =3 sites.
- Under certain conditions, the optimizer is unique and in Monge form.

## Abstract

We consider symmetric multi-marginal Kantorovich optimal transport problems on finite state spaces with uniform-marginal constraint. These problems consist of minimizing a linear objective function over a high-dimensional polytope, here referred to as Kantorovich polytope. The presented results are of split nature, computational and theoretical. Within the computational part only small numbers of marginals $N$ and marginal sites $\ell$ are considered. This restriction allows us to computationally determine all extreme points of the Kantorovich polytope and investigate how many of them are in compliance with the in optimal transport typical Monge ansatz. Singling out the results for $\ell=3$ discretization points and pairwise symmetric cost functions enables us to visually compare Kantorovich's to Monge's ansatz space for a varying number of marginals. Finally we present a necessary support-condition for optimizers which is inspired by the insights the said model problem on three sites provided. This result is not limited to the case of $\ell=3$ sites and applies to symmetric pair-costs whose diagonal entries lie above a cost-specific threshold. In case $N$ and $\ell$ display certain relationships the discussed condition provides an optimizer in Monge-form and implies its uniqueness as a solution of the considered Kantorovich optimal transport problem.