# $L^1$-Monge problem in metric spaces possibly with branching geodesics

## TL;DR

This paper proves the existence of optimal transport maps in certain metric spaces with branching geodesics, extending classical results to more complex geometries like normed spaces and Hilbert geometries.

## Contribution

It establishes the existence of optimal transport maps in metric spaces with branching geodesics, broadening the scope of optimal transport theory.

## Key findings

- Optimal transport maps exist in some metric spaces with branching geodesics.
- Results apply to normed spaces and Hilbert geometries.
- The existence holds when the first marginal is absolutely continuous.

## Abstract

In this paper, we consider the Monge optimal transport problem with distance cost. We prove that in some metric spaces, possibly with many branching geodesics, an optimal transport map exists if the first marginal is absolutely continuous. The result is applicable to normed spaces and Hilbert geometries.