# Convex integrals of molecules in Lipschitz-free spaces

## TL;DR

This paper introduces convex integrals of molecules in Lipschitz-free spaces, linking them to optimal transport and extremal structures, and characterizes extreme points when the underlying metric space is uniformly discrete.

## Contribution

It defines convex integrals of molecules in Lipschitz-free spaces and explores their properties, including their relation to Radon measures and extremal points.

## Key findings

- Convex integrals are determined by cyclical monotonicity.
- Under finiteness conditions, convex integrals coincide with measure-induced elements.
- All elements are convex series of molecules when the space is uniformly discrete.

## Abstract

We introduce convex integrals of molecules in Lipschitz-free spaces $\mathcal{F}(M)$ as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these elements are determined by cyclical monotonicity of their supports, and that under certain finiteness conditions they agree with elements of $\mathcal{F}(M)$ that are induced by Radon measures on $M$, or that can be decomposed into positive and negative parts. We also show that convex integrals differ in general from convex series of molecules. Finally, we present some standalone results regarding extensions of Lipschitz functions which, combined with the above, yield applications to the extremal structure of $\mathcal{F}(M)$. In particular, we show that all elements of $\mathcal{F}(M)$ are convex series of molecules when $M$ is uniformly discrete and identify all extreme points of the unit ball of $\mathcal{F}(M)$ in that case.