Nonexpansive map

Summary

A nonexpansive map between metric spaces is a map that never increases distances: $d(f(x), f(y)) \le d(x, y)$. These are exactly the maps that are Lipschitz with constant $1$; they are also called short maps or metric maps. They include isometries and contraction mappings. In fixed point theory they are the borderline case where the Banach fixed point theorem no longer applies and the geometry of the space matters.