# Supports and extreme points in Lipschitz-free spaces

### Ramón J. Aliaga

Universidad Politècnica de València, Valencia, Spain### Eva Pernecká

Czech Technical University, Prague, Czechia

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## Abstract

For a complete metric space $M$, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space $\mathcal{F}{M}$ are precisely the elementary molecules $(\delta(p)-\delta(q))/d(p,q)$ defined by pairs of points $p,q$ in $M$ such that the triangle inequality $d(p,q) < d(p,r)+d(q,r)$ is strict for any $r\in M$ different from $p$ and $q$. To this end, we show that the class of Lipschitz-free spaces over closed subsets of $M$ is closed under arbitrary intersections when $M$ has finite diameter, and that this allows a natural definition of the support of elements of $\mathcal{F}{M}$.

## Cite this article

Ramón J. Aliaga, Eva Pernecká, Supports and extreme points in Lipschitz-free spaces. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2073–2089

DOI 10.4171/RMI/1191