arxiv: 2604.26490 · v1 · submitted 2026-04-29 · 🧮 math.MG

Recognition: unknown

Directional curvature and medial axis

Adam Bia{\l}o\.zyt, Dominik Bysiewicz, Maciej P. Denkowski

Pith reviewed 2026-05-07 12:43 UTC · model grok-4.3

classification 🧮 math.MG

keywords medial axisdirectional curvaturecamber directionssuperquadraticitydefinable setssingularitiesplane case

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The pith

A criterion using directional curvature in camber directions characterizes when the medial axis reaches the singularities of a definable closed set.

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

The paper seeks to characterize the points in the intersection of the closure of the medial axis with the set X itself. These points mark where the medial axis touches X and correspond to its singularities. Assuming X is definable in a polynomially bounded structure tames the geometry enough to obtain a general criterion that generalizes the earlier notion of superquadraticity. The authors introduce directional curvature measured along naturally chosen camber directions and avoid any smoothness requirement on X. This completes the full description of the plane case.

Core claim

We obtain a general criterion based on a generalisation of the notion of superquadraticity by introducing a notion of directional curvature in some naturally chosen camber directions. This allows us in particular to complete the study of the plane case for characterising the points of the set of singularities given by the intersection of the closure of the medial axis with X.

What carries the argument

Directional curvature measured in naturally chosen camber directions, serving as the generalization of superquadraticity that detects medial-axis singularities without any smoothness assumption on X.

Load-bearing premise

The set X is definable in a polynomially bounded structure.

What would settle it

A concrete closed set X definable in a polynomially bounded structure together with an explicit point in the closure of its medial axis that lies in X, yet fails to satisfy the directional-curvature criterion in the chosen camber directions.

Figures

Figures reproduced from arXiv: 2604.26490 by Adam Bia{\l}o\.zyt, Dominik Bysiewicz, Maciej P. Denkowski.

**Figure 1.** Figure 1: Unlike the parabola, the curve y = |x| 3/2 is C 1 -smooth but not C 2 -smooth at the origin. Notation. We shall simplify the notation by putting X(1) := Sng1X, X(2) := (Reg1X ∩ Sng2X). Reaching Problem Characterize the points of the set MX ∩ X = view at source ↗

**Figure 2.** Figure 2: The medial axis of the curve y = x 3/2 , x ≥ 0 has a ‘non-obvious’ behaviour as a result of the curve’s superquadraticity. Up to now, only the case of subsets of R 2 has been thoroughly understood; see [6] and [11]; we complete this study in Sections 3 and 4. In the general case, we have the following Tangent Cone Criterion from [5] (see also [2]) applicable only at singular points: if the Peano tangent co… view at source ↗

**Figure 3.** Figure 3: If the X is not horizontally filled, then naturally the origin is achievable by MX, for any X satisfying (∗). horizontal direction prevents MX from approaching 0 from the tangent directions ( cf. 0 ∈ MX1 , whereas 0 ∈/ MX2 ), which is key in the later results. To define things precisely, let B(0, 2r) be the disk of radius 2r > 0, centered at 0. Thanks to (∗), we can take the radius r small enough to get B(… view at source ↗

read the original abstract

The medial axis $M_X$ of a closed set $X\subset \mathbb{R}^n$ is the set of points from the ambient space that admit more than one closest point in $X$. We study the problem of reaching the singularities, i.e. of characterising the points of the set $\overline{M_X}\cap X$. In order to tame the geometry, we assume that $X$ is definable in a polynomially bounded structure and obtain a general criterion based on a generalisation of the notion of superquadraticity previously introduced by Birbrair and Denkowski for $C^1$-smooth hypersurfaces and extended to any codimension by Bia{\l}o\.zyt. We do not require any smoothness as we achieve our goal by introducing a notion of directional curvature in some naturally chosen camber directions. This allows us in particular to complete the study of the plane case.

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 gives a criterion for medial axis points hitting the set X by defining directional curvature along camber directions for definable sets in polynomially bounded structures, and it finishes the plane case as a corollary.

read the letter

The core advance is a workable criterion for points in the closure of the medial axis that lie in X itself. The authors do this for closed sets definable in polynomially bounded o-minimal structures by introducing directional curvature measured in selected camber directions. This extends the earlier superquadraticity condition from smooth hypersurfaces to non-smooth sets of any codimension without assuming differentiability anywhere. The plane case then drops out directly as a special instance, which closes that particular question under the tameness hypothesis. That is the concrete new piece of work. The approach is straightforward once the directional curvature is set up, and the abstract frames the construction as a direct generalization of the authors' prior results and Birbrair-Denkowski. The tameness assumption is stated up front and used to keep the geometry under control, which is honest about the scope. One limitation is that the definability requirement excludes many sets that arise in applications outside o-minimal geometry, so the criterion stays inside a restricted class. The choice of camber directions also needs to be verified as truly natural rather than tailored to make the proof go through, though nothing in the description suggests circularity. The paper is aimed at specialists in geometric measure theory and computational geometry who already work with medial axes and singularities. A reader who needs to check when the skeleton touches the boundary in the plane under tame assumptions will get a usable test. It is narrow enough that it will not interest a broad audience, but the gap it fills is real. The work shows clear engagement with the existing literature on superquadraticity and medial axes, so it deserves a serious referee rather than a desk rejection. I would send it out for review.

Referee Report

0 major / 2 minor

Summary. The paper claims to characterize points in the closure of the medial axis M_X intersected with a closed set X in R^n by deriving a general criterion that generalizes the notion of superquadraticity. This is achieved by introducing directional curvature along naturally chosen camber directions, under the standing assumption that X is definable in a polynomially bounded o-minimal structure; the approach requires no smoothness and is used in particular to complete the study of the plane case.

Significance. If the derivation holds, the work supplies a concrete, non-smooth criterion for medial-axis singularities inside tame geometries, extending the superquadraticity results of Birbrair-Denkowski and Białozyt. The introduction of directional curvature as a new, geometrically natural object is a positive contribution that may find use in other questions involving singularities of distance functions or reach sets within o-minimal settings.

minor comments (2)

The abstract and introduction introduce 'camber directions' and 'directional curvature' without a preliminary informal description; adding one or two sentences that indicate how these objects are chosen from the geometry of the distance function would improve accessibility for readers outside the immediate subfield.
In the statement of the main criterion (presumably Theorem X or the central result in the section following the definitions), it would be helpful to include a short remark confirming that the new notion reduces to the classical superquadraticity condition when X is C^1-smooth, so that the generalization is manifestly consistent with prior work.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending minor revision. The referee summary accurately captures the main contributions, and we appreciate the recognition of the significance of directional curvature as a new geometric object in the o-minimal setting. Since the major comments section contains no specific points, we have no revisions to make in response to the report.

read point-by-point responses

Referee: The paper claims to characterize points in the closure of the medial axis M_X intersected with a closed set X in R^n by deriving a general criterion that generalizes the notion of superquadraticity. This is achieved by introducing directional curvature along naturally chosen camber directions, under the standing assumption that X is definable in a polynomially bounded o-minimal structure; the approach requires no smoothness and is used in particular to complete the study of the plane case.

Authors: We thank the referee for this precise summary of our results. The generalization of superquadraticity through directional curvature in camber directions is indeed the core of the paper, and we are pleased that its applicability to the plane case and lack of smoothness assumptions are highlighted. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states upfront the tameness assumption that X is definable in a polynomially bounded o-minimal structure. It then introduces a new notion of directional curvature in camber directions to generalize the prior superquadraticity concept (cited from Birbrair-Denkowski and Białożyt) and derives a criterion for points in closure(M_X) ∩ X. This new directional curvature supplies the independent step that handles the non-smooth case; the plane-case completion is listed as a corollary. Self-citations to overlapping-author prior work are present but not load-bearing, as the central derivation rests on the freshly defined curvature rather than reducing to a fit, self-definition, or unverified self-citation chain. The argument is self-contained against the explicit tameness hypothesis and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definability assumption to control geometry and on the new directional curvature definition; no free parameters or external data fitting are mentioned.

axioms (1)

domain assumption X is definable in a polynomially bounded structure
Invoked to tame the geometry and obtain the general criterion.

invented entities (1)

directional curvature no independent evidence
purpose: To characterize points in the closure of the medial axis intersected with X
New notion introduced along camber directions; no independent evidence provided beyond the paper's criterion.

pith-pipeline@v0.9.0 · 5456 in / 1136 out tokens · 29344 ms · 2026-05-07T12:43:23.665625+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 4 canonical work pages · 1 internal anchor

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Birbrair, M

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1979