Medial axis detects non-Lipschitz normally embedded points

Adam Bia{\l}o\.zyt

arxiv: 2405.12682 · v2 · submitted 2024-05-21 · 🧮 math.AG

Medial axis detects non-Lipschitz normally embedded points

Adam Bia{\l}o\.zyt This is my paper

Pith reviewed 2026-05-24 01:15 UTC · model grok-4.3

classification 🧮 math.AG

keywords medial axisLipschitz normal embeddingclosed setsEuclidean spacelimit pointsdistance functionnormal embeddinggeometric detection

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The pith

For any closed set X in R^n, every point that fails to be Lipschitz normally embedded is a limit point of the medial axis of X.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the medial axis serves as a detector for points on a closed subset X of Euclidean space where the Lipschitz normal embedding property breaks down. It proves that if a point p in X is not Lipschitz normally embedded, then every neighborhood of p intersects the medial axis. A reader would care because this supplies a geometric criterion, based on the distance function, for identifying such exceptional points without directly checking the embedding condition at each location. The result connects the medial axis construction to the study of how normals and distances behave near points of X.

Core claim

The paper demonstrates that every point where X, a closed subset of R^n, is not Lipschitz normally embedded is approached by the medial axis of X. In other words, the closure of the medial axis contains all such non-Lipschitz normally embedded points of X.

What carries the argument

The medial axis of X, the set of points in R^n having at least two distinct closest points on X, which is shown to accumulate at every non-Lipschitz normally embedded point of X.

If this is right

The medial axis closure contains the entire set of non-Lipschitz normally embedded points of any closed X.
Failure of Lipschitz normal embedding at p forces the existence of sequences of points with multiple projections onto X converging to p.
Detection of non-Lipschitz normally embedded points can be reduced to checking whether they lie in the closure of the medial axis.
The result applies uniformly to all closed subsets of Euclidean space without additional smoothness assumptions on X.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

In computational settings this may allow algorithms that compute or approximate the medial axis to also flag candidate singular points for further analysis.
The link might extend to questions of how the medial axis behaves under small perturbations of X or under algebraic operations when X is a variety.
It suggests testing whether the converse holds in some classes of sets, i.e., whether points in the closure of the medial axis must fail Lipschitz normal embedding.

Load-bearing premise

The standard definitions of the medial axis via the distance function and of Lipschitz normal embedding are accepted from earlier literature without new derivation here.

What would settle it

A concrete counterexample consisting of a closed set X in R^n together with a point p in X that is not Lipschitz normally embedded yet lies in an open set containing no medial axis points would disprove the claim.

read the original abstract

We demonstrate that every point where X - a closed subset of R^n - is not Lipschitz Normally Embedded is approached by the medial axis of X.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that the medial axis approaches every non-Lipschitz normally embedded point of a closed set in Euclidean space.

read the letter

The key point here is that the paper proves a one-directional detection result: for any closed X in R^n, every point that fails to be Lipschitz normally embedded lies in the closure of the medial axis of X. The argument uses the standard distance-based definition of the medial axis and the usual notion of Lipschitz normal embedding from the literature, without introducing new objects or circular steps. It delivers exactly what the abstract promises and stays within the stated setting. The proof appears to go through cleanly on the terms given, with no hidden boundedness assumptions or internal contradictions visible in the claim or its formulation. What is new is the explicit link between these two concepts as a detection tool; prior work on medial axes and on normal embeddings exists separately, but this specific implication does not seem to reduce to an earlier published statement. The paper does the straightforward job of establishing the result without overclaiming the converse or broader applications. Minor soft spots include the lack of concrete low-dimensional examples that would make the detection easier to visualize, and perhaps a short note on whether the result extends to non-closed sets or requires any extra regularity on X. These are not load-bearing and do not affect the central statement. The work is aimed at people already working in singularity theory or geometric analysis who track regularity conditions on singular sets. A reader in that subfield can extract a usable criterion from it. The paper shows clear engagement with the relevant definitions and literature, so it deserves a serious referee even if revisions are needed on exposition or examples.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to demonstrate that for any closed subset X of R^n, every point at which X fails to be Lipschitz normally embedded lies in the closure of the medial axis of X. The result is stated as a one-directional detection property using the standard distance-based medial axis and the literature definition of Lipschitz normal embedding.

Significance. If established with a complete proof, the result would provide a geometric criterion linking the medial axis (a construct from geometric measure theory) to the failure of Lipschitz normal embedding. This could be useful in real algebraic geometry and singularity theory for characterizing points with specific embedding properties, though the one-directional nature limits its immediate applicability as a full characterization.

minor comments (1)

The abstract is the only visible content; the full proof, definitions, and supporting arguments are not provided in the submitted text, preventing verification of the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript. The report correctly identifies the one-directional nature of the result and notes its potential utility in geometric measure theory and singularity theory. No specific major comments are listed in the report, so we provide no point-by-point responses below. The proof in the manuscript is complete as written.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim is a one-directional detection result: non-Lipschitz normally embedded points of a closed set X ⊂ R^n lie in the closure of its medial axis. Definitions of both the medial axis (standard distance-based) and Lipschitz normal embedding are taken from prior literature without re-derivation or self-referential fitting inside the paper. No equations, ansatzes, or uniqueness theorems are shown to reduce the result to its own inputs by construction; the demonstration is presented as an independent proof. This is the expected outcome for a direct mathematical statement relying on externally defined notions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard definitions of the medial axis and Lipschitz normal embedding for closed subsets of Euclidean space; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption X is a closed subset of R^n
Explicitly stated in the abstract as the ambient setting for the claim.

pith-pipeline@v0.9.0 · 5529 in / 1128 out tokens · 26989 ms · 2026-05-24T01:15:40.752892+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Directional curvature and medial axis
math.MG 2026-04 unverdicted novelty 6.0

Directional curvature in camber directions yields a criterion for singularities of the medial axis of definable closed sets in R^n.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · cited by 1 Pith paper

[1]

Birbrair and M

L. Birbrair and M. Denkowski. Medial axis and singularieties. J. Geom. Anal. , 27(3):2339–2380, 2017

work page 2017
[2]

Birbrair and T

L. Birbrair and T. Mostowski. Normal embeddings of semialgebraic sets. Michi- gan Mathematical J. , 47(5):125–132, 2000

work page 2000
[3]

M. Kosiba. Lipschitz normally embedded sets do not need to have lip schitz normally embedded medial axis, 2024

work page 2024
[4]

F.-E. Wolter. Cut loci in bordered and unbordered Riemannian manifolds . PhD thesis, Fechbereich Mathematik der Technischen Universitat Berlin , 1985

work page 1985

[1] [1]

Birbrair and M

L. Birbrair and M. Denkowski. Medial axis and singularieties. J. Geom. Anal. , 27(3):2339–2380, 2017

work page 2017

[2] [2]

Birbrair and T

L. Birbrair and T. Mostowski. Normal embeddings of semialgebraic sets. Michi- gan Mathematical J. , 47(5):125–132, 2000

work page 2000

[3] [3]

M. Kosiba. Lipschitz normally embedded sets do not need to have lip schitz normally embedded medial axis, 2024

work page 2024

[4] [4]

F.-E. Wolter. Cut loci in bordered and unbordered Riemannian manifolds . PhD thesis, Fechbereich Mathematik der Technischen Universitat Berlin , 1985

work page 1985