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arxiv: 2211.02124 · v2 · submitted 2022-11-03 · 🧮 math.GT

The Number of Singularities in the Intersections of Convex Planar Translates

Cameron Strachan This is my paper

Pith reviewed 2026-05-24 10:27 UTC · model grok-4.3

classification 🧮 math.GT
keywords convex bodiestranslatesintersectionssingularitiesboundaryplanar geometrystrict convexity
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The pith

The intersection of n contributing translates of a strictly convex smooth planar body has exactly n singular points on its boundary.

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

The paper proves that for a strictly convex and smooth convex body phi in the Euclidean plane, the intersection of any n translates has exactly n singular points on its boundary provided the intersection has nonempty interior and each translate is essential. This means that omitting any single translate strictly enlarges the intersection. The count is shown to be sharp because dropping strict convexity, smoothness, nonempty interior, or the contribution condition allows the number of singularities to differ from n. The result therefore gives a topological or geometric invariant that ties the number of essential translates directly to the number of corners on the resulting convex set.

Core claim

Let phi be a strictly convex, smooth convex body in the Euclidean plane. If the intersection of n translates of phi has nonempty interior and every translate contributes to the intersection, then that intersection has exactly n points of singularity along its boundary. The result is sharp: removing any hypothesis allows counterexamples with a different number of singularities.

What carries the argument

Singular points on the boundary of the intersection, which are the locations where the supporting line is contributed by two distinct translates and the boundary fails to be locally C1.

If this is right

  • The count of singularities equals the number of essential translates under the stated hypotheses.
  • Dropping strict convexity or smoothness permits intersections with more or fewer than n singularities.
  • When some translates do not contribute, the singularity count can fall below n.
  • When the intersection has empty interior the singularity count is no longer controlled by n.

Where Pith is reading between the lines

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

  • The result may extend to counting curvature jumps or changes in supporting lines rather than only corners.
  • Similar counting arguments could apply to Minkowski sums or other operations on convex bodies.
  • The n-to-n relation suggests a possible Euler-characteristic or degree argument that equates the number of active bodies to the number of transition points.

Load-bearing premise

Every translate must contribute to the intersection, so that removing any one enlarges the set.

What would settle it

An explicit set of n translates of a strictly convex smooth body whose intersection has positive area, each translate is essential, yet the boundary contains a number of singular points different from n.

Figures

Figures reproduced from arXiv: 2211.02124 by Cameron Strachan.

Figure 1
Figure 1. Figure 1: Demonstrating that a closed ball is a convex set [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Removing a boundary point of a closed ball remains to be a convex set [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Removing a boundary point of a square can produce a non-convex set [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Examples of convex sets [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Examples of non-convex sets 8 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A counterexample to Theorem 2.11 if we remove the assumption of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Examples of polytopes 2.4 Convex Bodies We have now gone over the basics of convexity. The three different classifications of convexity we derived gives us a strong foundation of understanding these sets. I would like to finish off this section with a discussion of a special class of convex sets, the class of convex bodies. This class of convex shapes is quite well-known, and has been studied to great dept… view at source ↗
Figure 8
Figure 8. Figure 8: An example of a convex shape with 4 singularities, all of them having infinitely many support [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: If a boundary point has two normal’s, it also has all normal’s in between [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: An ideal example showing Theorem 3.1 I developed Theorem 3.1 first by considering the intersection of finitely many balls, with non-empty interior, and with no redundancies in the intersection; these shapes are know as ball polytopes. After proving this I proceeded to generalize the case to the largest class of convex bodies I could find. This then led to Theorem 3.1. 16 [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 11
Figure 11. Figure 11: Counterexample to Theorem 3.1 if we remove the assumption of translates [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Counterexample to Theorem 3.1 if we remove the assumption of strict convexity [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Counterexample to Theorem 3.1 if we remove the assumption of smoothness [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Counterexample to Theorem 3.1 if we remove the assumption of non-empty interior [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Counterexample to Theorem 3.1 if we remove the assumption that none of our translates are [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The intersection of two balls in E 3 3.3 Counterexample to Theorem 3.1 in higher dimensions With all the assumptions in Theorem 3.1 justified, we can now discuss why this theorem does not hold in higher dimensions. Consider the intersection of two balls in E 3 , which is depicted in [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Two translates of φ cannot have a boundary intersection with opposite normal vectors Lemma 4.1 tells us that if two translates intersect with non-empty interior, and the translates are not equal to each other; then where the boundary of these two translates intersect, we must have non-parallel support planes for each translate. The following lemma shows that these points, where the boundaries of two trans… view at source ↗
Figure 18
Figure 18. Figure 18: The intersection of the boundary of two translates will produce a singular point [PITH_FULL_IMAGE:figures/full_fig_p022_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: If a boundary point belongs to the boundary of only one translate, then it will be regular [PITH_FULL_IMAGE:figures/full_fig_p022_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Examples of chords in various convex bodies [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Examples of chord functions in various convex bodies [PITH_FULL_IMAGE:figures/full_fig_p024_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The limit of the chords κi being smaller than the chord at the limit, κ With this diagram we see that the end points of the converging sequence of chords of κi , will converge to the end point of limi→∞ κi , which will be contained in κ. However, from our diagram we see that there must exists a k ∈ N sufficiently large such that the line segment between the end point of κ and the end point of κk is not co… view at source ↗
Figure 23
Figure 23. Figure 23: The limit of the chords κi , being larger than the chord of the limit κ 24 [PITH_FULL_IMAGE:figures/full_fig_p024_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: The convex hull of the two chords κ and κ 0 Since these chords have endpoints that belong to the boundary of a convex body, there must be two segments of the boundary of φ that connects the endpoints of each chord of maximum length. Since the parallelogram must be contained in φ, these boundary segments must not pierce the interior of the parallelogram. I claim that the boundary segments also cannot be ou… view at source ↗
Figure 25
Figure 25. Figure 25: If the bd(φ) is outside the parallelogram Since the boundary segments connecting the end points of our chords can neither go outside nor inside the parallelogram, it must be the case that the boundary segment of φ connecting the end points, is the line segment connecting the end points. But this is a contradiction as φ was assumed to be strictly convex and hence cannot contain a line segment in its bounda… view at source ↗
Figure 26
Figure 26. Figure 26: The convex hull of κ m0 and κ max It can be easily shown that since the chords around the chord κ m0 , have strictly smaller length than κ m0 , there must be an end point of a chord in φ that belongs to the interior of P. This is a contradiction: as the end point of this curve is a boundary point of φ, which because φ is a convex body we know that it has a support plane at this end point, but any support … view at source ↗
Figure 27
Figure 27. Figure 27: Example of θ ⊥w for a strictly convex, convex body 26 [PITH_FULL_IMAGE:figures/full_fig_p026_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Orange chords of φ1 being mapped to the blue chords of φ2 [PITH_FULL_IMAGE:figures/full_fig_p027_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Example of Lemma 4.9 for an ellipse [PITH_FULL_IMAGE:figures/full_fig_p029_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: A picture to visualize our labels [PITH_FULL_IMAGE:figures/full_fig_p029_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: A picture to visualize more labels As we saw from our interlude in chord theory φ2 = τ (φ1) will be contained in the hyperplanes Hw and H−w and will support φ2 at the point h 0 w and h 0 −w respectively [PITH_FULL_IMAGE:figures/full_fig_p029_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: The left side of φ2 cannot intersect the left side of φ1 So bd(φ2) may only intersect the right side of bd(φ1) which implies that m(Γφ1 (bd(φ1) ∩ φ2)) ≤ π Since bd(φ1) ∩ φ2 is contained in the right side of φ1. It is easily seen that bd(φ1) ∩ φ2 will be a closed boundary segment of φ1 which implies the Gauss image of this boundary segment will be closed as well. This together with the fact that a closed s… view at source ↗
Figure 33
Figure 33. Figure 33: Part of an edge of Φ0 cannot be the only thing outside of φn+1 π, which would imply the boundary segment that passes outside of φn+1 has measure less than π. This leads to a contradiction as this would imply that m(Γφi (bd(φi) \ {φn+1})) < π Which is a contradiction to Lemma 4.9. 2.b) Suppose sing(Φ0 ) * sing(Φ) then there exists a singularity si ∈ sing(Φ0 ) such that si ∈/ sing(Φ). I claim there must be … view at source ↗
Figure 34
Figure 34. Figure 34: φn+1 cutting off the singularity si , implies |sing(Φ)| = n + 1 However, we see that φn+1 has removed exactly one singularity, but has only added two (each one corresponding to the two edges φn+1 intersects) this is a contradiction as this would imply that |sing(Φ)| = |sing(Φ0 ∩ φn+1)| = n + 1 which contradicts our assumption |sing(Φ)| ≥ n + 2. Thus, there must be at least two singularities si , sj ∈ sing… view at source ↗
Figure 35
Figure 35. Figure 35: φn+1 cutting off two singularities of Φ0 contradiction, or it will not cut of the singularity, a contradiction). Also note that si and sj cant be adjacent singularities as this would imply either φn+1 cuts off the entire edge in between si and sj , or we have φn+1 intersecting the edge in between them twice (which is easily seen to be a contradiction using Lemma 4.9). With this stated we see there are at … view at source ↗
Figure 36
Figure 36. Figure 36: bdφn+1 intersects bd(Φ0 ) in at least 4 points Since all four of these points belong to the boundary of a strictly convex, smooth, convex body, φn+1, and are distinct, all four Gauss images of these points will be distinct: Γφn+1 (ni) 6= Γφn+1 (nj ) For all i 6= j ∈ {1, 2, 3, 4}. Since we have 4 distinct points, ordered clockwise on S 1 it must be the case that either: ∠ cw(Γφn+1 (n1), Γφn+1 (n2)) < π or … view at source ↗
Figure 37
Figure 37. Figure 37: The Gauss images of our 4 points, n1, n2, n3 and n4 [PITH_FULL_IMAGE:figures/full_fig_p033_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: Φ0 , φi and φn+1 Imagine starting from the point n1 in the diagram above, and going up along the boundary of φn+1 (which is initially outside of both φi and Φ0 ). Let w be the second point of intersection in bd(φn+1) ∩ bd(φi) (the first being n1). Since Φ0 ⊆ φi , as we move along the boundary of φn+1 we must make contact with w before n2 which implies that: ∠ cw(Γφn+1 (n1), Γφn+1 (w)) ≤ ∠ cw(Γφn+1 (n1), Γ… view at source ↗
read the original abstract

This purpose of this paper is to prove the following result: let phi be a strictly convex, smooth, convex body in the Euclidean plane, if the intersection of n translates of phi has a non-empty interior, and all of the translates contribute to the intersection, then the intersection of these n translates will have exactly n points of singularity along its boundary. Furthermore this result is sharp, in the sense that, removing any one of the assumptions from our statement will render the result unable to hold in general.

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 manuscript proves that if φ is a strictly convex and smooth convex body in the Euclidean plane, then the intersection of n translates of φ that has nonempty interior and to which each translate contributes has exactly n singular points on its boundary. The result is shown to be sharp via explicit counterexamples obtained by dropping any one of the hypotheses.

Significance. If the argument holds, the result gives a precise, sharp count on the number of boundary singularities arising from cyclic arrangements of smooth arcs contributed by each translate. This supplies a basic structural fact in convex geometry that may be useful for analyzing boundaries of intersections of convex sets.

minor comments (1)
  1. The abstract and introduction repeat the phrase 'convex body' after already stating strict convexity and smoothness; a single clean statement of the hypotheses would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending acceptance. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a direct counting theorem for singularities on the boundary of an intersection of n strictly convex smooth translates under the stated hypotheses. The abstract and claim contain no equations, fitted parameters, self-citations, or ansatzes. The argument relies on the geometric consequences of strict convexity (boundary arcs from each translate) and the 'all contribute' condition (ensuring each supplies a positive-length arc), which together force exactly n junction points by elementary topology of closed curves. No step reduces the conclusion to a redefinition or prior fit by the same authors. This is a self-contained geometric result with no load-bearing circular elements.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definition of strict convexity and smoothness for a convex body in R^2 together with the geometric notion of a translate contributing to an intersection. No free parameters or invented entities are introduced.

axioms (2)
  • domain assumption A strictly convex smooth body in the plane has a well-defined boundary with unique supporting lines at each point.
    Invoked by the statement that phi is strictly convex and smooth; this is standard in convex geometry.
  • domain assumption The intersection having nonempty interior and every translate contributing are independent geometric conditions that can be checked separately.
    Explicitly required in the theorem statement.

pith-pipeline@v0.9.0 · 5598 in / 1289 out tokens · 30925 ms · 2026-05-24T10:27:33.304750+00:00 · methodology

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Reference graph

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