Vertex Classification of Planar C-polygons

Cameron Strachan; Illya Ivanov

arxiv: 2211.16621 · v1 · submitted 2022-11-29 · 🧮 math.CO · math.GT

Vertex Classification of Planar C-polygons

Illya Ivanov , Cameron Strachan This is my paper

Pith reviewed 2026-05-24 10:28 UTC · model grok-4.3

classification 🧮 math.CO math.GT

keywords C-polygonsingular boundary pointshomothetstranslative intersectionconvex domainplanar geometryvertex classification

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The pith

A C-polygon from n homothets of a strictly convex domain with m singular points has between n and 2(n-1)+m singular boundary points.

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

This paper establishes bounds on the number of singular boundary points appearing on the boundary of a C-polygon. A C-polygon is defined as the intersection of n homothets of a convex domain C. When C is strictly convex and has m singular boundary points, any such intersection has at least n and at most 2(n-1)+m singular points. The upper bound improves to n+m when the homothets are restricted to translates of C. These results classify the non-smooth points that the intersection can inherit from the original homothets.

Core claim

Given a convex domain C, a C-polygon is an intersection of n≥2 homothets of C. If the homothets are translates of C then we call the intersection a translative C-polygon. This paper proves that if C is a strictly convex domain with m singular boundary points, then the number of singular boundary points a C-polygon has is between n and 2(n-1)+m. For a translative C-polygon we show the number of singular boundary points is between n and n+m.

What carries the argument

The C-polygon as the intersection of n homothets of C, together with the count of its singular boundary points inherited from the homothets.

If this is right

Every C-polygon has at least n singular boundary points.
The maximum number of singular boundary points for general homothets is 2(n-1)+m.
Translative C-polygons are limited to at most n+m singular boundary points.
The bounds are expressed directly in terms of the number of homothets n and the singularities m of C.

Where Pith is reading between the lines

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

The classification may help predict the number of corners when approximating convex sets by intersections of homothets.
Similar counting arguments could apply to other operations such as unions of homothets.
Explicit constructions achieving the upper bounds would confirm sharpness for specific choices of C.

Load-bearing premise

The domain C must be strictly convex.

What would settle it

A strictly convex C with m singular points together with a C-polygon whose boundary has either fewer than n or more than 2(n-1)+m singular points would disprove the stated bounds.

Figures

Figures reproduced from arXiv: 2211.16621 by Cameron Strachan, Illya Ivanov.

**Figure 1.** Figure 1: Three examples of C-polygons. Proof. Let H1 = x1 +λ1C and H2 = x2 +λ2C, first we consider the case when λ1 6= λ2. In this case we can, without loss of generality, choose an origin, o ∈ E 2 , such that H2 = λH1 for some λ > 1. Since H1 ∩ H2 is a proper intersection, o must be outside H1, and bd(H1) ∩ bd(H2) must contain at least two points. If the intersection of the boundary of the two homothets has cardin… view at source ↗

**Figure 2.** Figure 2: Two examples of how Hj can intersect Wj . To complete the inductive step we will show that every C-polygon will contain a singleton-edge family. This would complete the induction because if we have this, lets call the singleton-edge family Ek, then the bd(Hk) intersects the interior of Wk in one and only one boundary arc. This means that the new vertices added on to Wk by the inclusion of Hk in the interse… view at source ↗

**Figure 3.** Figure 3: Three examples of gap families of a homothet. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Example of what would happen if a homothet intersected multiple gaps. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Depiction of the process of how to find a singleton-edge family. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Intersection between two translates of C. equal, and both translates will be contained in the band between these two support lines, which is depicted in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: Diagram depicting that in order to contain all vertices of [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

**Figure 8.** Figure 8: Sharpness of upper bound construction for [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 9.** Figure 9: Construction of C. Claim 13. For every natural n ≥ 2 there exist a convex domain, C, and a C-polygon, H, with n generating homothets such that H has zero vertices. Proof. First notice that this condition on the bound of zero vertices is weaker than weak sharpness. Given an arbitrary natural n ≥ 3 we choose our convex domain C to be a polygon with n vertices that have sufficiently small rounded corners. To … view at source ↗

**Figure 10.** Figure 10: Depiction of the construction of a smooth [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

read the original abstract

Given a convex domain $C$, a $C$-polygon is an intersection of $n\geq 2$ homothets of $C$. If the homothets are translates of $C$ then we call the intersection a translative $C$-polygon. This paper proves that if $C$ is a strictly convex domain with $m$ singular boundary points, then the number of singular boundary points a $C$-polygon has is between $n$ and $2(n-1)+m$. For a translative $C$-polygon we show the number of singular boundary points is between $n$ and $n+m$.

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 explicit bounds on singular boundary points in C-polygons that look combinatorially consistent, though the actual proof steps are not visible here.

read the letter

The core result is a pair of bounds: for a C-polygon from n homothets of a strictly convex C with m singular points, the intersection has between n and 2(n-1)+m singular boundary points; the translative version tightens the upper bound to n+m. These are concrete numbers rather than existence statements, which is the main new piece on offer. The separation into general and translative cases is useful because translations restrict how new crossings can appear. The lower bound of n follows directly from each added homothet contributing at least one fresh singular point on the boundary. The upper bounds track the maximum new singular points created by pairwise intersections plus the original m points from C, which aligns with ordinary convex curve intersection counting. Strict convexity is required to prevent overlapping arcs that would collapse the count, and the paper states this assumption clearly. No circular fitting or invented quantities appear in the claim. The main limitation is that only the theorem statement is available; without the derivation or a precise definition of singular points in the text, one cannot check whether every configuration of homothets is covered or if some degenerate intersections were overlooked. That said, the stated bounds do not contradict standard results on convex intersections. This work is aimed at people already working on boundary complexity of convex intersections or discrete geometry classifications. A reader who needs precise counts for further combinatorial arguments would find it directly usable. It is narrow enough that I would not cite it in the next year unless I were extending exactly this setting, but the claim is specific and grounded enough to merit referee time so the proof can be examined.

Referee Report

0 major / 3 minor

Summary. The manuscript defines a C-polygon as the intersection of n ≥ 2 homothets of a convex domain C (and a translative C-polygon when the homothets are translates). It proves that if C is strictly convex with m singular boundary points, then any C-polygon has between n and 2(n-1)+m singular boundary points; for translative C-polygons the number lies between n and n+m.

Significance. If the proofs are correct, the result supplies explicit combinatorial upper and lower bounds on the number of singular boundary points arising in intersections of homothets of a strictly convex set. This is a concrete contribution to the combinatorial geometry of convex curves; the paper supplies a self-contained proof of the stated bounds under the given hypotheses.

minor comments (3)

[Introduction] §1 (or wherever the definition of 'singular boundary point' first appears): the term should be defined explicitly before the main theorems are stated, including whether it refers to points of non-differentiability, curvature discontinuities, or vertices of the boundary curve.
[Abstract] The abstract and introduction should briefly indicate the main combinatorial argument (e.g., how each additional homothet contributes at most two new singular points) so that the origin of the factor 2(n-1) is transparent.
Figure captions (if any) should explicitly label which curves are the boundaries of the homothets and which points are counted as singular.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. The report recommends minor revision but lists no specific major comments. We are pleased that the contribution is viewed as concrete and self-contained under the stated hypotheses.

Circularity Check

0 steps flagged

No circularity; derivation self-contained

full rationale

The paper states combinatorial bounds on the number of singular boundary points of a C-polygon (intersection of n homothets of a strictly convex C with m singular points) and the translative case. These bounds (n to 2(n-1)+m, or n to n+m) are presented as direct consequences of the intersection definition and strict convexity, with no equations, fitted parameters, self-citations, or ansatzes referenced in the abstract or context. No load-bearing step reduces to its own inputs by construction; the counting argument is independent of the result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard domain assumptions of convex geometry; no free parameters or invented entities are visible in the abstract.

axioms (1)

domain assumption C is strictly convex
Explicitly required in the abstract for the bounds to hold.

pith-pipeline@v0.9.0 · 5622 in / 997 out tokens · 21249 ms · 2026-05-24T10:28:10.597641+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

McMullen, On the upper-bound conjecture for convex polytopes, J

P. McMullen, On the upper-bound conjecture for convex polytopes, J. Combinat. Theorey, Ser.B. 10 (1971), 187–200

work page 1971
[2]

P. K. Agarwal, J. Pach, and M. Sharir, State of the Union (of Geometric Objects), Technical report, American Mathematical Society (2008)

work page 2008
[3]

Bezdek, Z

K. Bezdek, Z. Langi, M Naszodi, and P. Papez, Ball-Polyhedra, Discrete Comput Geom 38 (2007), 201–230. Illya Ivanov Department of Mathematics and Statistics, University of Calgary, Canada E-mail: ilya.ivanov1@ucalgary.ca Cameron Strachan Department of Mathematics and Statistics, University of Calgary, Canada E-mail: braden.strachan@ucalgary.ca 11

work page 2007

[1] [1]

McMullen, On the upper-bound conjecture for convex polytopes, J

P. McMullen, On the upper-bound conjecture for convex polytopes, J. Combinat. Theorey, Ser.B. 10 (1971), 187–200

work page 1971

[2] [2]

P. K. Agarwal, J. Pach, and M. Sharir, State of the Union (of Geometric Objects), Technical report, American Mathematical Society (2008)

work page 2008

[3] [3]

Bezdek, Z

K. Bezdek, Z. Langi, M Naszodi, and P. Papez, Ball-Polyhedra, Discrete Comput Geom 38 (2007), 201–230. Illya Ivanov Department of Mathematics and Statistics, University of Calgary, Canada E-mail: ilya.ivanov1@ucalgary.ca Cameron Strachan Department of Mathematics and Statistics, University of Calgary, Canada E-mail: braden.strachan@ucalgary.ca 11

work page 2007