# Kantorovich Duality and Optimal Transport Problems on Magnetic Graphs

## TL;DR

This paper develops duality theory for Lipschitz and Arens-Eells function spaces on magnetic graphs, extending optimal transport concepts to discrete magnetic structures and characterizing their geometric properties.

## Contribution

It introduces magnetic Lipschitz and Arens-Eells spaces, establishes duality, and characterizes their extreme points and quotient structures, adapting optimal transport tools to magnetic graphs.

## Key findings

- Duality between magnetic Lipschitz and Arens-Eells spaces established.
- Extreme points of the magnetic Lipschitz space unit ball characterized.
- Magnetic Arens-Eells space identified as a quotient of classical Arens-Eells space.

## Abstract

We consider Lipschitz- and Arens-Eells-type function spaces constructed for magnetic graphs, which are adapted to this setting from the area of optimal transport on discrete spaces. After establishing the duality between these spaces, we prove a characterization of the extreme points of the unit ball in the magnetic Lipschitz space as well as a result identifying the magnetic Arens-Eells space as a quotient of the classical Arens-Eells space of an associated classical graph called the lift graph.