# On an identification of the Lipschitz-free spaces over subsets of $\mathbb{R}^{n}$

## TL;DR

This paper develops a method to identify Lipschitz-free spaces over open convex subsets of Euclidean space, showing they can be represented as quotients of certain L^1 spaces, advancing understanding of their structure.

## Contribution

The paper introduces a new approach to identify Lipschitz-free spaces over convex subsets of ^n, linking them to quotients of L^1 spaces, which was not previously established.

## Key findings

- Identified Lipschitz-free spaces over convex subsets of ^n.
- Established that these spaces are quotients of L^1(U;^n).
- Provided a method for explicit identification of these spaces.

## Abstract

In this short note, we develop a method for identifying the spaces $Lip_{0}(U)$ for every nonempty open convex $U$ of $\mathbb{R}^{n}$ and $n\in\mathbb{N}$. Moreover, we show that $\mathcal{F}(U)$ is identified with a quotient of $L^{1}(U;\mathbb{R}^{n})$.