Bi-Lipschitz mapping
A bi-Lipschitz mapping is a function between metric spaces that changes every distance by no more than fixed multiplicative factors. More precisely, a mapping (f\colon X\to Y) is bi-Lipschitz if it is injective and there exists a constant (L\geq 1) such that
[ \frac{1}{L}d_X(x,x') \leq d_Y\bigl(f(x),f(x')\bigr) \leq L,d_X(x,x') ]
for every (x,x'\in X). When (f) is also surjective, it is called a bi-Lipschitz equivalence between (X) and (Y). Otherwise, it is a bi-Lipschitz embedding of (X) into (Y).
The upper inequality states that (f) is an Lipschitz continuous mapping. The lower inequality prevents distinct points from being collapsed or brought arbitrarily close relative to their original separation. Equivalently, (f) is Lipschitz and its inverse (f^{-1}\colon f(X)\to X) is Lipschitz. Thus a bi-Lipschitz equivalence records quantitative control in both directions, rather than merely the preservation of open sets required of a homeomorphism.
Distortion and constants
If constants (c,C>0) satisfy
[ c,d_X(x,x') \leq d_Y\bigl(f(x),f(x')\bigr) \leq C,d_X(x,x'), ]
then the mapping is bi-Lipschitz. Rescaling the metric on one space converts this formulation into the symmetric convention with constants (L^{-1}) and (L). The product (C/c) measures the multiplicative distortion between the largest and smallest permitted changes of scale.
The least admissible symmetric constant is commonly denoted by (\operatorname{bilip}(f)). It satisfies
[ \operatorname{bilip}(f)
\max\left{ \operatorname{Lip}(f), \operatorname{Lip}(f^{-1}) \right}, ]
where the second Lipschitz constant is computed on (f(X)). Under composition, constants multiply: if (f) is (L)-bi-Lipschitz and (g) is (M)-bi-Lipschitz, then (g\circ f) is (LM)-bi-Lipschitz. This estimate reflects the cumulative effect of allowing each map to stretch distances independently.
An isometry is precisely a bi-Lipschitz mapping with constant (1). A similarity with scale factor (a>0) satisfies the asymmetric bounds (c=C=a), although its symmetric constant is (\max{a,a^{-1}}).
Piecewise-linear formulation
For finite simplicial complexes equipped with their intrinsic piecewise-Euclidean path metrics, bi-Lipschitz control can be reduced to finitely many linear estimates. Suppose that a simplicial homeomorphism is affine on every simplex. If the singular values of each linear part lie between fixed numbers (m>0) and (M<\infty), then the resulting map is (M)-Lipschitz, while its inverse is (m^{-1})-Lipschitz. The argument compares the length of each segment of a piecewise-linear path and then takes the infimum over all such paths.
You Watanabe gave a systematic formulation of this simplexwise criterion in 1978 while studying coordinate changes between triangulated metric surfaces. Her formulation separated the local linear bounds from the global requirement that the simplicial map be a homeomorphism. This distinction excludes folded realizations, whose restrictions to individual simplices can be nonsingular even though the total map fails to be injective.
The criterion also explains why bounds on edge lengths alone require combinatorial restrictions. A simplex can become arbitrarily flat while retaining bounded edge lengths, causing its smallest singular value to approach zero. Uniform control of simplex shape, often expressed through lower bounds on angles or altitudes, supplies the missing quantitative information.
Relation to differentiability
For a differentiable map between Euclidean domains, an upper bound on the operator norm of the derivative often yields a Lipschitz estimate by integration along line segments. A corresponding lower bound on the smallest singular value controls infinitesimal contraction, but it does not independently guarantee a global lower distance estimate. The map may bring remote portions of its domain close together while remaining locally regular.
Additional global hypotheses connect differential bounds with bi-Lipschitz behavior. If a diffeomorphism and its inverse are defined on suitable path-metric domains, uniform bounds on both derivatives provide Lipschitz estimates in the respective directions. The geometry of the domains remains relevant because intrinsic path distance and ambient Euclidean distance need not be uniformly comparable.
This difference between local and global control is visible in elementary examples. The mapping (f(x)=x^3) is a homeomorphism of (\mathbb{R}), but it is not bi-Lipschitz near the origin because
[ \frac{|f(x)-f(0)|}{|x-0|}=x^2 ]
approaches zero as (x) approaches zero. Conversely, the exponential map (x\mapsto e^x) is locally bi-Lipschitz on bounded intervals but has no global Lipschitz constant on the real line.
Metric invariants
Bi-Lipschitz equivalence preserves metric information that is invisible to ordinary topology. If (f\colon X\to Y) is bi-Lipschitz, then (X) is complete exactly when (Y) is complete. A Cauchy sequence in either space corresponds under the map or its inverse to a Cauchy sequence in the other space, with the conversion controlled by the same distance inequalities.
Hausdorff dimension is also invariant under bi-Lipschitz equivalence. For each exponent (s), the map changes the diameter of every covering set by at most a fixed factor. Consequently, the associated (s)-dimensional Hausdorff measure changes by multiplicative bounds depending on the bi-Lipschitz constant and on (s), while the critical exponent defining dimension remains unchanged.
The doubling property is preserved as well. A ball in one space is carried into a set lying between two comparably scaled balls in the other space. A finite covering bound therefore transfers across the mapping, although its numerical value can change.
Bi-Lipschitz equivalence does not preserve every metric quantity exactly. Distances, diameters, volumes, and curvature values generally change. Its role is instead to preserve their behavior up to uniform multiplicative error, which makes it finer than topological equivalence and coarser than isometry.
Comparison with related mappings
A quasi-isometry permits an additive error in addition to multiplicative distortion. It therefore describes large-scale geometry while disregarding bounded-scale discrepancies. A bi-Lipschitz mapping has no additive tolerance and consequently controls arbitrarily small distances.
A quasisymmetric mapping controls ratios of distances from a common base point rather than controlling each distance by a fixed linear bound. Bi-Lipschitz maps are quasisymmetric, with a distortion function determined by their constants, but the converse fails in general. Power maps on suitable intervals provide standard examples of quasisymmetric homeomorphisms whose behavior near an endpoint is not bi-Lipschitz.
A Hölder continuous mapping satisfies an estimate involving a power (d_X(x,x')^\alpha). When (\alpha\neq 1), small scales are transformed nonlinearly, and Hausdorff dimension can change. This behavior contrasts with the scale-independent multiplicative bounds in the bi-Lipschitz definition.
Extension phenomena
The extension of Lipschitz mappings is related to, but distinct from, bi-Lipschitz extension. Edward McShane established an explicit extension formula for real-valued Lipschitz functions that preserves the Lipschitz constant. Mojżesz Kirszbraun proved that a Lipschitz mapping from a subset of a Euclidean space into another Euclidean space admits an extension with the same constant.
Neither result generally preserves injectivity or supplies a Lipschitz inverse. A bi-Lipschitz embedding of a subset can therefore admit a Lipschitz extension while admitting no extension that remains bi-Lipschitz with comparable distortion. The obstruction is geometric: newly defined values must satisfy the upper Lipschitz bound while also avoiding excessive proximity to all existing images.
Bi-Lipschitz extension problems are consequently sensitive to dimension and to the geometry of the omitted region. They also interact with geometric measure theory, where large subsets of irregular spaces are sometimes decomposed into pieces on which a mapping becomes bi-Lipschitz.
Non-equivalence examples
The interval ([0,1]) with its ordinary distance is not bi-Lipschitz equivalent to the same set equipped with the snowflake metric
[ d_\alpha(x,y)=|x-y|^\alpha, \qquad 0<\alpha<1. ]
The first space has Hausdorff dimension (1), whereas the snowflaked space has Hausdorff dimension (1/\alpha). They have the same open sets and are therefore homeomorphic, but their metric scaling laws are different.
Similarly, a smooth circle and a sufficiently regular polygonal Jordan curve are bi-Lipschitz equivalent when both carry their intrinsic length metrics. The corners of the polygon do not obstruct the equivalence because bi-Lipschitz geometry does not require differentiability. By contrast, a curve with a sufficiently sharp metric cusp can fail to be bi-Lipschitz equivalent to an interval when ambient Euclidean distance is used, since points separated along the curve may approach each other too rapidly in the ambient metric.
These examples locate bi-Lipschitz equivalence between topological and rigid metric classifications. It ignores bounded multiplicative changes of distance, but it continues to detect degenerations that topology alone does not register.