Bi-Lipschitz Functional Embedding

• Bi-Lipschitz functional embedding is a mapping that preserves distances up to uniform multiplicative constants, ensuring that metric structures are accurately maintained.
• It leverages local-to-global techniques such as Christ cube and Whitney decompositions to construct embeddings that scale from fine geometric details to overall structure.
• The framework facilitates quantitative comparisons with canonical spaces and has practical applications in sub-Riemannian geometry, data science, and operator analysis.

A bi-Lipschitz functional embedding is a map between metric spaces that preserves distances up to a uniform multiplicative constant, with a particular focus on function-valued mappings that encode the original space into a Banach or Hilbert space. Such embeddings are central in geometric analysis, metric geometry, and applications spanning analysis on metric spaces, data science, signal processing, geometric group theory, and theoretical computer science. The existence and construction of bi-Lipschitz embeddings elucidate the geometric structure of a space and enable quantitative comparisons with canonical geometries, such as Euclidean or Banach spaces.

1. Fundamental Definition and Notions

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A map $f : X \to Y$ is called bi-Lipschitz if there exist constants $0 < \ell \leq L < \infty$ such that for all $x, x' \in X$,

$$\ell \, d_X(x, x') \leq d_Y(f(x), f(x')) \leq L \, d_X(x, x').$$

The minimal $L / \ell$ is called the distortion of $f$. Functional embeddings of the form $x \mapsto d(x, -)$ into $L^2(\mu)$ or related mappings are canonical tools for representing the metric structure of $X$ in a linear space.

A bi-Lipschitz functional embedding refers to such a mapping—often into a function space (e.g., $L^2$), Hilbert space, or a finite-dimensional Euclidean space—preserving the metric up to uniform multiplicative constants.

2. Structural Characterization via Local Embeddability

The characterization of when a metric space can be bi-Lipschitz embedded into Euclidean space involves aspects of both global and local geometry. In (Seo, 2011), the following is established for a complete, doubling, uniformly perfect metric space $(X,d)$:

• $X$ bi-Lipschitz embeds into some Euclidean space if and only if:
  1. $X$ carries a doubling measure,
  2. There exists a closed subset $Y \subset X$ that bi-Lipschitz embeds into $\mathbb{R}^{M_1}$,
  3. The complement $\Omega = X \setminus Y$ admits uniformly "Christ–local" bi-Lipschitz embeddings into $\mathbb{R}^{M_2}$, meaning that the local geometric structure (decomposed dyadically via Christ cubes) permits consistent embeddings at all scales.

Key technical tools include Christ's dyadic decomposition, McShane's Lipschitz extension, and a Whitney-type decomposition to pass from local to global embeddability. The proof of the co-Lipschitz condition—ensuring the mapping does not collapse distances—is constructed by analyzing "large scale" (far-apart points) versus "local" (nearby points within the Whitney structure) cases, using cutoff functions and coloring methods to coordinate local embeddings.

3. Canonical Distance Function Embeddings

A central construction (see (Movahedi-Lankarani et al., 13 Jan 2025)) is the distance function embedding $\iota_d: x \mapsto d(x, -)$, viewed as an element in $L^2(\mu)$. This encoding is always Lipschitz: $$\|\iota_d(x) - \iota_d(x')\|_{L^2(\mu)}^2 = \int_X |d(x, z) - d(x', z)|^2\, d\mu(z),$$ but is bi-Lipschitz if and only if there exists $c > 0$ such that for all $x \neq x'$,

$$\|\iota_d(x) - \iota_d(x')\|_{L^2(\mu)} \geq c\, d(x, x').$$

Sufficient conditions for the existence of a bi-Lipschitz finite-dimensional reduction from this embedding are uniform point separation and lower Lipschitz properties of $\iota_d$ or of constructed "convolution" metrics. Radial projection and analysis of the Hilbert–Schmidt operator $T_d$ induced by the metric provide operator-theoretic frameworks for establishing such embeddings.

4. Decompositional Techniques: Whitney, Christ Cubes, and Local–Global Patching

Constructing global bi-Lipschitz embeddings often necessitates local-to-global synthesis. (Seo, 2011) develops a multiscale decomposition:

• Christ cubes: Analogous to dyadic cubes in Euclidean geometry, these partition a doubling metric space at all scales, allowing controlled local analysis.
• Whitney-type decompositions: The complement of a "good" set is covered by cubes $Q$ such that $$\operatorname{diam}(Q) \simeq \operatorname{dist}(Q, Y)$$, defining the appropriate scale for "locality".
• Coloring and cutoff functions: These are used to assign disjoint target coordinates and to localize the support of the embedding map, ensuring that nearby cubes are distinguished in the embedding and that the patchwork retains bi-Lipschitz control.

Local bi-Lipschitz embeddings, once constructed and patched, admit extension to the full space via extension theorems such as McShane's.

5. Applications to Singular and Sub-Riemannian Structures

The framework is applied to sub-Riemannian manifolds, particularly the Grushin plane. The Grushin distribution is generated by $X_1 = \partial_x$ and $X_2 = x \partial_y$, with Carnot–Carathéodory metric inherited from these vector fields: $$ds^2 = dx^2 + \frac{dy^2}{x^2};\quad\text{length}(\gamma) = \int_0^1 \sqrt{x'(t)^2 + \frac{y'(t)^2}{x(t)^2}}\, dt.$$ By separating the singular set $Y = \{x = 0\}$ (where the metric degenerates) and embedding $Y$ via snowflaked metric techniques (Assouad's theorem), and applying the Whitney decomposition to the regular set $\Omega$, the global embedding of the Grushin plane is achieved. These results yield the first known example of a singular sub-Riemannian manifold admitting a bi-Lipschitz embedding into Euclidean space; similar arguments apply to higher-dimensional spaces of Grushin type.

6. Implications and Broader Context

The methods and results described in (Seo, 2011, Movahedi-Lankarani et al., 13 Jan 2025), and related works provide a robust toolkit for the geometric analysis of metric spaces:

• Quantitative Embeddability: Membership of a class of spaces in the family of bi-Lipschitz embeddable spaces can now be characterized by local metric and analytic conditions (doubling, uniform perfectness, local embedding properties).
• Classification and Examples: Classical or singular spaces (sub-Riemannian, fractal, or spaces with singularities) can be classified according to their embeddability, with concrete criteria and construction methods.
• Operator-Theoretic and Functional Perspectives: The viewpoint of embedding via function-valued mappings ($x \mapsto d(x, -)$) connects the intrinsic geometry of the space with structures in functional and operator analysis, facilitating finite-dimensional reductions and practical constructions.

A plausible implication is that these frameworks may be extendable to wider classes of metric spaces of analytic or algorithmic interest, and the classification of embedding obstructions and sharp constants remains a prime direction for future research.