Complete metric spaces with property (Z) are length spaces

Abstract

A metric space M is length when every two points have an approximate midpoint, while M has property ( Z ) when for every x , y ∈ M there exists z such that d ( x , z ) + d ( z , y ) − d ( x , y ) is arbitrarily smaller than d ( x , z ) and d ( z , y ) . Answering a problem posed by García-Lirola, Procházka and Rueda Zoca, we prove that every complete metric space with property ( Z ) is length. As a consequence, we get that for the Lipschitz-free space of a metric space a number of geometric properties are all equivalent, like the Daugavet property, the diameter-2 property, the absence of strongly exposed points or the absence of preserved extreme points.