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arxiv: 2003.03690 · v4 · submitted 2020-03-08 · 🧮 math.MG

On the singular points approached by the medial axis

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

Pith reviewed 2026-05-24 14:46 UTC · model grok-4.3

classification 🧮 math.MG
keywords superquadracitymedial axissingular pointsregularity conditionC1 smoothR^n subsetsgeometric analysis
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The pith

Superquadracity of a set in R^n implies non-empty intersection with the closure of its medial axis.

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

The paper extends the notion of superquadracity for subsets of R^n originally defined by Birbrair and Denkowski. Its main theorem shows that this property is related to the set intersecting the closure of its medial axis. It further examines non C1 smooth points of the set. A reader would care if this provides a way to detect or characterize singular points using the medial axis. The work focuses on developing this regularity condition to establish the geometric relation.

Core claim

The central theorem establishes the relation between superquadracity and the non-empty intersection of the set and the closure of its medial axis. The investigation also concerns non C1 smooth points of the set.

What carries the argument

Superquadracity as a regularity condition on subsets of R^n that relates to the behavior of the medial axis.

If this is right

  • If a set is superquadratic then it intersects the closure of its medial axis.
  • Non C1 smooth points are approached by the medial axis under this condition.
  • The theorem holds for subsets of R^n.

Where Pith is reading between the lines

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

  • This could be applied to computational geometry for finding singularities.
  • It may connect to other regularity notions in real algebraic geometry.
  • Examples in low dimensions could be used to test the theorem's sharpness.

Load-bearing premise

Superquadracity as defined is the suitable regularity condition for the intersection property to hold with the medial axis closure.

What would settle it

Constructing a superquadratic set in R^n that avoids intersecting the closure of its medial axis would disprove the theorem.

Figures

Figures reproduced from arXiv: 2003.03690 by Adam Bia{\l}o\.zyt.

Figure 1
Figure 1. Figure 1: A graph from the Example 2.9 Corollary 2.8 gives full information about MX ∩ Reg1X when X is a subset of a plane. The problem becomes more complicated in higher dimensions, and Theorem 2.6 cannot be reversed. Example 2.9. First observe, that for a cone C = {z 2 = x 2 + y 2} and a point p with x, y coordinates equal to zero, m(p) is a full circle parallel to the XY plane, and MC ∩ {x = y = 0} ∩ C = {0}. Nex… view at source ↗
Figure 2
Figure 2. Figure 2: The surface from the Example 3.3. the tangent cone at the origin is equal to R 2×{0}. Therefore, the lower limit of tangent cones cannot be a superset of the tangent cone at the origin, and consequently, the medial axis of X reaches the origin. Sadly, an analogous inequality involving the tangent cone and the upper Kuratowski limit does not ensure an approach of the medial axis. One can see it easily from … view at source ↗
Figure 3
Figure 3. Figure 3: The set X = {(z − p3 x 2 + y 2 )(x 2 + y 2 ) = 0} from Example 3.9. We are ready now to show the final result on the non-marginal C 1 singularities approached by the medial axis. Corollary 3.8. Assume that X is a closed definable set, x ∈ X. Let Γ ⊂ X be a topological manifold of dimension dimx X, and assume that x ∈ Γ\MX. Then, there exists U ∈ V(x) such that Γ ∩ U = X ∩ U, and, in particular, X is locall… view at source ↗
read the original abstract

This paper develops the notion of superquadracity defined by L.Birbrair and M.Denkowski for subsets of R^n. In this regard, the main theorem of the paper establishes the relation between the superquadracity and non-empty intersection of the set and the closure of its medial axis. The further investigation concerns non C1 smooth points of the set.

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.

Referee Report

0 major / 1 minor

Summary. The paper develops the Birbrair-Denkowski notion of superquadracity for subsets of R^n. Its main theorem relates this property to the non-empty intersection of the set with the closure of its medial axis. Additional investigation addresses non-C^1 smooth points of the set.

Significance. If the central theorem holds, the work supplies a regularity condition linking superquadracity to medial-axis geometry, which may be useful for analyzing singularities in geometric sets. The manuscript builds directly on prior definitions without introducing new free parameters, ad-hoc axioms, or invented entities.

minor comments (1)
  1. The abstract states that a main theorem exists but supplies neither the precise statement of the theorem, the relevant definitions, nor any proof outline, which limits immediate assessment of the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for the concise summary of its content and potential significance. No specific major comments were provided in the report, so we have no points to address individually at this stage. We remain available to clarify any aspects of the main theorem relating superquadracity to the closure of the medial axis or the discussion of non-C^1 points.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends the externally defined notion of superquadracity from Birbrair and Denkowski (cited as prior work) and proves a relation to the medial axis closure. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The central claim rests on an independent regularity condition imported from outside the paper rather than on quantities or theorems constructed internally by the present authors. This is the normal case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified because only the abstract is available; the paper relies on a prior definition whose details are not reproduced here.

pith-pipeline@v0.9.0 · 5571 in / 979 out tokens · 15991 ms · 2026-05-24T14:46:49.635903+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages · 1 internal anchor

  1. [1]

    The tangent cone, the dimension and the frontier of a medial axis

    A. Bia lo˙ zyt. The tangent cone, the dimension and the frontier of a medial axis . 2021. arXiv: 2003.10436 [math.MG]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  2. [2]

    Geometric characterizations of C1 man- ifolds in Euclidean spaces by tangent cones

    F. Bigolin and G. Greco. “Geometric characterizations of C1 man- ifolds in Euclidean spaces by tangent cones”. In: J Math. Anal. Appl. 396.1 (2012), pp. 145–163. issn: 0022-247X

    work page 2012

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    Medial axis and Singularieties

    L. Birbrair and M. Denkowski. “Medial axis and Singularieties”. In: J. Geom. Anal. 27.3 (2017), pp. 2339–2380

    work page 2017

  4. [4]

    Denkowska and M

    A. Denkowska and M. Denkowski. The Kuratowski convergence of medial axes and conflict sets. 2016. arXiv: 1602.05422 [math.MG]

    work page arXiv 2016

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    Tangent cones and regularity of real hypersurfaces

    M. Ghomi and R. Howard. “Tangent cones and regularity of real hypersurfaces”. In: J. Reine Angew. Math. 2014 (May 2010)

    work page 2014

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    Real Algebraic Manifolds

    J. Nash. “Real Algebraic Manifolds”. In: Annals of Mathematics 56.3 (1952), pp. 405–421

    work page 1952

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    On the structure of sets with positive reach

    J. Rataj and L. Zaj´ ıcek. “On the structure of sets with positive reach”. In: Math. Nachr. 290 (2016), pp. 1806–1829

    work page 2016