Complete metric spaces with property (Z) are length metric spaces

TL;DR

This paper proves that all complete metric spaces with property (Z) are length spaces, providing answers to open questions about the structure of Lipschitz-free Banach spaces related to these metric spaces.

Contribution

It establishes that complete metric spaces with property (Z) are length spaces, resolving previously posed questions in the field.

Findings

Complete metric spaces with property (Z) are length spaces.

Answers to open questions about Lipschitz-free Banach spaces.

Clarifies the structure of certain metric spaces.

Abstract

We prove that every complete metric space with property (Z) is a length space. These answers questions posed by Garc\'{i}a-Lirola, Proch\'{a}zka and Rueda Zoca, and by Becerra Guerrero, L\'{o}pez-P\'{e}rez and Rueda Zoca, related to the structure of Lipschitz-free Banach spaces of metric spaces.