BiLipschitz

BiLipschitz, often written bi-Lipschitz, describes a strong form of distance preservation between metric spaces. A map f between metric spaces (X, d_X) and (Y, d_Y) is bi-Lipschitz if there exists a constant C ≥ 1 such that for all x, x' in X, (1/C) d_X(x, x') ≤ d_Y(f(x), f(x')) ≤ C d_X(x, x').

Equivalently, f is Lipschitz with constant C and has a Lipschitz inverse on its image. In particular, f is injective and f(X) is a metric subspace of Y. Bi-Lipschitz maps are homeomorphisms onto their image, preserving the topology and Cauchy sequences up to a controlled distortion.

Bi-Lipschitz equivalence of metric spaces means there exists a bijection f: X → Y that is bi-Lipschitz. This provides a strong notion of geometric sameness: distances are preserved up to a uniform multiplicative distortion, and features such as dimension, local structure, and many geometric properties are retained under such maps. In contrast, mere Lipschitz maps may distort distances, and quasi-isometries relax the condition further by allowing additive distortions and non-injectivity.

Examples and non-examples help illustrate the concept. The identity map on any metric space is bi-Lipschitz with constant C = 1. In Euclidean space, invertible linear maps A: R^n → R^n are bi-Lipschitz, with constant max(||A||, ||A^{-1}||). By contrast, a map that collapses distances or has unbounded distortion, such as x ↦ x^3 on the real line, is not bi-Lipschitz on all of R.

In practice, bi-Lipschitz equivalence is central to geometric analysis, rigidity results, and the study of manifolds and spaces up to controlled distortion.