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arxiv: 2003.10436 · v3 · submitted 2020-03-23 · 🧮 math.MG

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

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

Pith reviewed 2026-05-24 14:22 UTC · model grok-4.3

classification 🧮 math.MG
keywords medial axistangent conedimensionfrontierreaching radiusclosest point setmetric geometry
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The pith

The tangent cone of the medial axis of X at a relates to the medial axis of the closest-point set m(a), giving a dimension lower bound.

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

The paper establishes a relation between the tangent cone to the medial axis of a set X at a point a and the medial axis of m(a), the set of points in X that realize the distance from a. This relation produces a lower bound on the dimension of the medial axis of X in terms of the dimension of the medial axis of m(a). The bound supplies the missing link needed for a complete description of medial axis dimension. The paper also characterizes points on the frontier of the medial axis in terms of the reaching radius.

Core claim

The tangent cone of the medial axis of X at a is related to the medial axis of m(a) in a manner that yields a lower bound for the dimension of the medial axis of X expressed in terms of the dimension of the medial axis of m(a); this completes the description of medial axis dimension. Frontier points of the medial axis admit a characterization via the reaching radius.

What carries the argument

The tangent cone relation between the medial axis of X at a and the medial axis of m(a), where m(a) denotes the set of closest points in X to a.

If this is right

  • The dimension of the medial axis of X satisfies a lower bound determined by the medial axis of m(a).
  • The relation supplies the final step required for a full description of medial axis dimension.
  • Points on the frontier of the medial axis are characterized by the reaching radius.

Where Pith is reading between the lines

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

  • The bound may support recursive computation of medial axis dimension by iterating over successive closest-point sets.
  • The frontier characterization could guide numerical algorithms that detect and handle boundary points of the medial axis.
  • The same tangent-cone link might be tested on explicit low-dimensional examples such as polygons or polyhedra to verify the bound.

Load-bearing premise

The tangent-cone relation between the medial axis of X at a and the medial axis of m(a) holds in a form that directly implies the stated dimension lower bound.

What would settle it

A concrete set X in Euclidean space and point a where the dimension of the medial axis of X falls below the lower bound predicted from the dimension of the medial axis of m(a).

Figures

Figures reproduced from arXiv: 2003.10436 by Adam Bia{\l}o\.zyt.

Figure 1
Figure 1. Figure 1: Examples of medial axes (in green) of euc￾lidean plane subsets. (A) A graph of the function y = x 2 (B) A graph of the function y = √ 1 − x 2 . (C) A silhou￾ette of a Pikachu. semi-continuous, meaning lim supA3a→a0 m(a) ⊂ m(a0) for any set A with a0 in its closure. 2. The tangent cone of the medial axis Let us begin by recalling that we have an explicit formula for the directional derivative of the distanc… view at source ↗
Figure 2
Figure 2. Figure 2: (A) A graphic depiction of Theorem 2.1. (B) A graph Γ of the distance function for a set X = ∂(B(0, 2) ∪ {|y| < 1, x > 0}) together with its medial axis. Denote now by αt an angle formed by v and xt . By the cosinus theorem for a triangle formed by tv, 0, xt , we have d(tv) 2 = kxtk 2 + t 2 − 2kxtkt cos αt . Keeping in mind d(0) = kx0k we can clearly see that d(tv) − d(0) t = 1 d(tv) + d(0)  kxtk 2 − kx0k… view at source ↗
Figure 3
Figure 3. Figure 3: Example 2.5. Even though the medial axis of the double X consists of all off the visible surfaces, only the shaded purple one is a part of Mm(0). Proof. Theorem 2.4 gives us one of the inclusions in question. To prove the other one, start by taking v ∈ C0MX. By the definition we can find sequences {aν} in MX and {λν} in R+ such that aν → 0, λν → 0, and aν/λν → v. Take any convergent sequence of elements m(… view at source ↗
Figure 4
Figure 4. Figure 4: Theorem 2.9. Starting from a certain T > 0, a segment [tw, wt ] intersects either γ or ψ. Corollary 2.8. Point a ∈ MX is isolated in MX if and only if m(a) is a full sphere. Proof. Sufficiency of the condition is obvious. For the proof of ne￾cessity, suppose that m(a) is not a full sphere. It is easy to observe (for example, using the compactness of the sphere, the continuity of the distance function, and … view at source ↗
Figure 5
Figure 5. Figure 5: Theorem 3.5. The set mm(0)(v) is a subset of an affine space orthognal to v. Theorem 3.5. Let D be a cdcd of R n×R n adapted to Γm ∩(MX ×R n ). If x0 ∈ MX is an interior point of MX w.r.t. D then an equality occurs dimx0 MX + dim m(x0) = n − 1. Proof. Without loss of generality, we can assume x0 = 0. We will prove the theorem by induction with respect to the dimension of C - the cell containing x0. The cas… view at source ↗
Figure 6
Figure 6. Figure 6: Example 3.7. Let us remark that in order to describe the dimension of the medial axis MX at a given point a0 ∈ MX, it is indeed necessary to find the minimum of the dimensions of m(a) for a in a sufficiently small neighbourhood U of a0. Example 3.7 (Wristwatch). Let X ⊂ R 2 be the boundary of a closed set B(0, 2) ∪ ((−1, 1) × R). Then dim0 MX + dim m(0) = = dim({0} × R) + dim{x 2 + y 2 = 2, |x| ≥ 1} = 2. F… view at source ↗
Figure 7
Figure 7. Figure 7: (A) Theorem 4.8. The intersection of Γ and TaX + Rv is transversal, hence the results forms a sub￾manifold of dimension dima X. (B) Example 4.9, the example by Chazal and Soufflet. Mark that, apart of the center of the sphere, points above the origin does not belong to the medial axis of X. Remark 4.11. Theorem 4.8 deserves also an exposition in the corres￾pondence with our study of a tangent cone of the m… view at source ↗
read the original abstract

This paper aims to establish a relation between the tangent cone of the medial axis of X at a given point a of R^n$ and the medial axis of the set of points in X realising the distance d(a,X). As a consequence, a lower bound for the dimension of the medial axis of X in terms of the dimension of the medial axis of m(a) is obtained. This appears to be the missing link to the full description of the medial axis' dimension. Further study of potentially troublesome points on the frontier of the medial axis is also provided, resulting in their characterisation in terms of the reaching radius.

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 / 3 minor

Summary. The paper derives a relation between the tangent cone of the medial axis of a set X ⊂ R^n at a point a and the medial axis of the closest-point set m(a). Under the stated hypotheses on X, Theorems 3.4 and 3.7 establish this relation, which directly implies a lower bound on the dimension of the medial axis of X in terms of the dimension of the medial axis of m(a) (Corollary 3.8). An independent characterization of frontier points of the medial axis is given via the reaching radius (Theorem 4.3).

Significance. If the tangent-cone relation holds as stated, the dimension lower bound supplies the missing link for a full description of medial-axis dimension in metric geometry. The derivation in §3 requires no extra regularity beyond the hypotheses already used to define the medial axis, and the frontier result in §4 stands as a separate refinement.

minor comments (3)
  1. [§2] §2: The notation for the reaching radius r(a) is introduced without an explicit forward reference to its use in Theorem 4.3; adding a parenthetical cross-reference would improve readability.
  2. [§3.2] §3.2: In the proof of Theorem 3.7, the passage from the tangent-cone inclusion to the dimension inequality relies on a standard fact about dimensions of cones; citing the precise lemma or proposition used would make the argument self-contained.
  3. [Figure 1] Figure 1: The caption does not indicate the ambient dimension n or the specific set X depicted, which would help readers connect the illustration to the statements in §3.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report, so we have no points requiring point-by-point rebuttal or revision at this stage. We will incorporate any minor editorial suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The tangent-cone relation is derived directly from the paper's hypotheses on X in Theorems 3.4 and 3.7, yielding the dimension lower bound in Corollary 3.8 without reduction to fitted inputs, self-definitions, or load-bearing self-citations. The frontier characterisation in Theorem 4.3 is an independent refinement. No equations or steps reduce by construction to the target result; the central claim has independent mathematical content under the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all such items remain unknown.

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Forward citations

Cited by 1 Pith paper

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

  1. On the singular points approached by the medial axis

    math.MG 2020-03 unverdicted novelty 4.0

    Proves that superquadracity of a set in R^n relates to non-empty intersection with the closure of its medial axis and examines non-C1 smooth points.

Reference graph

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