Lipschitz Normal Embedding (LNE)

• Lipschitz Normal Embedding (LNE) is defined as the property where the intrinsic metric is uniformly bounded by the Euclidean distance, preventing metric detours.
• Key characterizations like the arc-criterion and test-curve methods are employed to establish metric equivalence in singular and analytic spaces.
• LNE plays a critical role in classifying singularities and links geometric, algebraic, and o-minimal theories, while presenting open challenges in higher-dimensional settings.

A subset $X$ of Euclidean space is Lipschitz normally embedded (LNE) if its intrinsic (inner) metric, measuring the length of shortest curves within $X$, is controlled up to a uniform constant by its extrinsic (outer) metric, given by ambient Euclidean distance. Specifically, for any $x, y \in X$, the distance along $X$ between $x$ and $y$ (computed as the infimum of lengths of rectifiable curves in $X$ joining $x$ to $y$) is bounded above by a constant times the ambient Euclidean distance between $x$ and $y$. This property captures the absence of metric "detours"—the geometry of $X$ does not force path-lengths to be arbitrarily larger than the ambient chord length, and is fundamental in geometric singularity theory, real algebraic, and analytic geometry.

1. Fundamental Definitions and Metric Equivalence

Let $X \subset \mathbb{R}^n$ be a closed or locally closed set, or more generally a germ at $0$. Equip $X$ with:

• Outer (ambient) metric: $d_\mathrm{out}(x, y) := \|x - y\|$ (Euclidean norm).
• Inner (intrinsic) metric: $$d_\mathrm{in}(x, y) := \inf \{ \operatorname{length}(\gamma) : \gamma \subset X \text{ rectifiable joining } x \text{ to } y \}$$.

By construction, $d_\mathrm{out}(x, y) \leq d_\mathrm{in}(x, y)$. The set $X$ is Lipschitz normally embedded if there exists $C \ge 1$ so that for all $x, y \in X$, $d_\mathrm{in}(x, y) \le C\, d_\mathrm{out}(x, y)$. The smallest such $C$ is called an LNE constant of $X$ (Fantini et al., 2022, Kerner et al., 2017, Neumann et al., 2015, Misev et al., 2018).

This definition extends to complex analytic sets using the Hermitian metric and to subanalytic and definable sets in o-minimal structures (Huynh et al., 23 Nov 2025, Mendes et al., 2021, Nguyen, 2021).

2. Arc-Criterion and Test-Curve Characterizations

A central tool for verifying the LNE property is the arc-criterion: A germ $(X, 0)$ is LNE if and only if for every pair of real analytic arcs $\gamma_1, \gamma_2$ emanating from $0$, the order of vanishing of the ambient distance $\|\gamma_1(t) - \gamma_2(t)\|$ equals the order of vanishing of the intrinsic metric $d_\mathrm{in}(\gamma_1(t), \gamma_2(t))$ (Mendes et al., 2021, Fantini et al., 2022, Fernandes et al., 2017, Kosiba, 15 Mar 2024).

For normal complex surface singularities, the Neumann–Pedersen–Pichon test-curve criterion gives a finite set of "test arcs" arising from generic projections and resolution graphs; LNE holds if and only if for every test arc:

• Multiplicity matching: principal preimage components have the same multiplicity as the plane arc.
• Contact-exponent matching: outer and inner contact exponents of principal preimage arcs coincide (Neumann et al., 2018, Neumann et al., 2015, Misev et al., 2018, Fantini et al., 2022).

This criterion unifies earlier necessary conditions from polar-curve theory and tangent cone reductions, and has analogs for semialgebraic and determinantal sets (Misev et al., 2018, Pedersen et al., 2016, Kerner et al., 2017).

3. Tangent Cones, Links, and Reduced Structures

If $X$ is LNE at $0$, its tangent cone $T_0 X$ (as defined via the spherical blow-up or by initial ideals in the complex/algebraic case) is also LNE and is reduced, meaning local branch multiplicities are 1 for every direction (Fernandes et al., 2017, Fantini et al., 2022). This is a necessary condition; however, the converse fails—sets with reduced and LNE tangent cone may still not be LNE themselves (e.g., certain subanalytic or complex analytic examples with isolated singularity) (Fernandes et al., 2017).

Links—small intersections of $X$ with spheres around the point—play a conical role: Mendes–Sampaio and Nguyen established that for subanalytic or definable sets, the LNE property of the germ is equivalent to uniform LNE of all links (termed LLNE) when the deleted link is connected (Mendes et al., 2021, Nguyen, 2021, Fantini et al., 2022). This strengthens the understanding that the metric geometry of singularities can be analyzed through the geometry of their links.

4. Algebraic and Analytic Examples: Minimality and Regularity Criteria

In complex surface singularity theory, LNE characterizes minimal rational singularities: The only rational surface singularities which are LNE are those whose fundamental cycle on the resolution is reduced (Neumann et al., 2015, Neumann et al., 2018). For curves, affine algebraic curves are LNE if and only if at each singular point the germ consists of finitely many smooth transverse branches, and at infinity the projective closure meets the hyperplane transversely at $d$ smooth points (where $d$ is the degree) (Costa et al., 2023).

For hypersurface germs given as images of finite analytic maps, LNE is equivalent to smoothness if the multiplicity matches the generic degree (i.e., $\operatorname{mult}(f) = \operatorname{gd}(f)$), with special rigidity holding for all finite corank-1 and injective map germs (Ballesteros et al., 29 Jul 2025).

In the field of determinantal varieties, all model strata—spaces of matrices of fixed rank, their closures, and many sections—are LNE, with explicit bilipschitz constants: the closure of rank-$r$ matrices in $m\times n$ space is LNE with constant $2\sqrt{2}$ (Kerner et al., 2017, Katz et al., 2016, Pedersen et al., 2016).

5. Geometric Constructions and Universality

Higher-dimensional examples reveal increased ambient complexity. In dimension four, LNE Hölder triangles exhibit universality: for any knot $K \subset S^3$, there exists an LNE Hölder triangle in $\mathbb{R}^4$ whose ambient bi-Lipschitz equivalence class encodes the knot type. Hence, ambient bi-Lipschitz geometry in dimension four comprises infinitely many inequivalent classes, contrasting with the ambient triviality in dimension three (Birbrair et al., 12 Oct 2024).

Medial axis geometry demonstrates that LNE of a set does not force LNE of its medial axis in dimensions $n \ge 3$; explicit semialgebraic counterexamples exist (Kosiba, 15 Mar 2024). However, in the plane, the equivalence between LNE of the set and its medial axis is restored for curve germs.

6. Invariance, Model Theory, and Rigidity Under Hölder Maps

Recent work on bi-$\alpha$-Hölder invariance in o-minimal structures shows that the LNE property, tangent cone dimension, and link homotopy are preserved under homeomorphisms with Hölder exponent exceeding a critical threshold $\alpha_0$ (depending on the definable geometry) (Huynh et al., 23 Nov 2025). For complex analytic germs, a bi-$\alpha$-Hölder equivalence to the Euclidean germ for $\alpha$ near 1 implies the germ is smooth.

LNE is also preserved under definable inner bi-Lipschitz homeomorphisms. For every closed connected definable subset $X \subset \mathbb{R}^p$, there exists a definable LNE model bi-Lipschitz for the inner metric. The one-point compactification theorem further reduces the global LNE problem for unbounded definable sets to the local compact theory on the sphere (Costa et al., 2023).

7. Open Problems and Research Directions

Current open questions center on combinatorial and metric classification of LNE singularities, especially in dimensions higher than two, and on finding finite-criterion analogs of arc and test-curve methods. The connection between Nash transform, jet schemes, and structural invariants with LNE remains unresolved. Clarifying when LNE of a singularity propagates through natural geometric operations (e.g., blowups, stratifications) remains a major challenge (Fantini et al., 2022).

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Lipschitz normal embedding occupies a central role in the metric classification of singularities, offering deep connections with topology, algebraic geometry, and o-minimality. It is characterized by a rich set of necessary and sufficient criteria, sharp invariants, and explicit model spaces. Its rigidity properties and open conjectures continue to motivate advances in the geometry of singular spaces.