Complete metric spaces with property \((Z)\) are length spaces
The authors call a complete metric space \(( M,d) \) a length space if every pair of points in \(M\) has an approximate midpoint (Definition 1.1). A pair of distinct points \(x,y\) in \(M\) is said to have property \((Z) \) if, for every \(\varepsilon >0\), there exists \(z\) such that \(d(x,z)+d(z,y)\leq d(x,y)+\varepsilon\max\{d(x,z),d(y,z)\}\), that is, \(d(x,z)+d(z,y)-d(x,y)\) is arbitrarily smaller than \(d(x,z)\) and \(d(y,z)\). A~metric space has property \((Z)\) if every pair of its distinct points has property \((Z)\). The main result, answering a question of \textit{L. García-Lirola} et al. [J. Math. Anal. Appl. 464, No. 1, 473--492 (2018; Zbl 1397.46008)], states that a complete metric space is a length space if and only if it has property \((Z)\).
scientific article; zbMATH DE number 7040409