[#P2882] Minimum distinct diagonal intersections in a convex polygon

A neutral schematic of the objects and relations in the statement.

Problem. For each integer \(n\ge 4\), let \(f(n)\) be the minimum, over all strictly convex planar \(n\)-gons, of the number of distinct points in the polygon's interior that lie on two or more diagonals; a point where several diagonals concur is counted once. Is \(f(n)\sim n^4/24\) as \(n\to\infty\)?

1Context

The objective compresses many diagonal crossings into shared points. Coordinate families, exact concurrence partitions, and lower-bound inequalities can be compared and reused as the best constant changes.

2Remarks

Remark 1. A diagonal joins two nonconsecutive vertices of the polygon.

Remark 2. A strictly convex polygon has distinct vertices in convex position and every interior angle strictly smaller than pi.

3What counts as a solution

Prove that f(n)/(n^4) tends to 1/24, or disprove the assertion by giving a different asymptotic bound separated from 1/24.
A disproof by constructions must specify convex coordinates and prove the claimed count or asymptotic count of distinct concurrence points; a proof must control arbitrary multiple concurrences.

1Status

Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow page has zero answers. Poonen and Rubinstein establish the n^4/24 leading term for regular polygons, while the checked searches found no proof that arbitrary convex polygons cannot create asymptotically more concurrence. Exact unresolved remainder: Prove that f(n)/(n^4) tends to 1/24, or disprove the assertion by giving a different asymptotic bound separated from 1/24. A disproof by constructions must specify convex coordinates and prove the claimed count or asymptotic count of distinct concurrence points; a proof must control arbitrary multiple concurrences.^[2][1][3]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow page has zero answers. Poonen and Rubinstein establish the n^4/24 leading term for regular polygons, while the checked searches found no proof that arbitrary convex polygons cannot create asymptotically more concurrence. Exact unresolved remainder: Prove that f(n)/(n^4) tends to 1/24, or disprove the assertion by giving a different asymptotic bound separated from 1/24. A disproof by constructions must specify convex coordinates and prove the claimed count or asymptotic count of distinct concurrence points; a proof must control arbitrary multiple concurrences.^[2][1][3] Relevance to this problem For Minimum distinct diagonal intersections in a convex polygon, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 1, Reference 3 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The MathOverflow page has zero answers. Poonen and Rubinstein establish the n^4/24 leading term for regular polygons, while the checked searches found no proof that arbitrary convex polygons cannot create asymptotically more concurrence. Exact unresolved remainder: Prove that f(n)/(n^4) tends to 1/24, or disprove the assertion by giving a different asymptotic bound separated from 1/24. A disproof by constructions must specify convex coordinates and prove the claimed count or asymptotic count of distinct concurrence points; a proof must control arbitrary multiple concurrences. Open full record →
	Trace	Reported
attempt · Route 1 Prove that f(n)/(n^4) tends to 1/24, or disprove the assertion by giving a different asymptotic bound separated from 1/24. A disproof by constructions must specify convex coordinates and prove the claimed count or asymptotic count of distinct concurrence points; a proof must control arbitrary multiple concurrences.^[1] Relevance to this problem For Minimum distinct diagonal intersections in a convex polygon, record convex-polygon-diagonal-intersection-asymptotic-next-route-20260801 (“Complete the stated acceptance conditions”) documents a concrete method, search boundary, or failed route. The record states: Prove that f(n)/(n^4) tends to 1/24, or disprove the assertion by giving a different asymptotic bound separated from 1/24. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work against the displayed statement and preserve every hypothesis and quantifier. Prove that f(n)/(n^4) tends to 1/24, or disprove the assertion by giving a different asymptotic bound separated from 1/24. A disproof by constructions must specify convex coordinates and prove the claimed count or asymptotic count of distinct concurrence points; a proof must control arbitrary multiple concurrences. Any computation must retain a replayable witness and a matching exclusion or completeness certificate. Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Poonen and Rubinstein establish the n^4/24 leading term for regular polygons, while the checked searches found no proof that arbitrary convex polygons cannot create asymptotically more concurrence.

On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for question 212361; comments provide small values and candidate point sets only.
Poonen and Rubinstein, SIAM Journal on Discrete Mathematics 11 (1998), compute regular-polygon intersections and give n^4/24+O(n^3), which supplies the current upper construction for f(n).
OEIS A230281 records the least known values only for small n. Exact-title and asymptotic searches did not locate a later matching lower bound.
A TheoremDB search for convex polygon diagonal concurrency, distinct interior intersections, and the n^4/24 asymptotic found no duplicate.
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. For n=4 one has f(4)=1, and a regular hexagon has 13 distinct interior diagonal-intersection points.

Computational notes

The source reports exact experiments for vertices on the parabola (i,i^2) through n=200; those data suggest substantial finite-size effects without deciding the leading constant.

2See also

Conway’s thrackle conjecturediscrete geometry
Borsuk’s conjecture in four dimensionsdiscrete geometry
Completing a line arrangement to triangular bounded cellsdiscrete geometry

How to cite

TheoremDB contributors, “Minimum distinct diagonal intersections in a convex polygon,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/convex-polygon-diagonal-intersection-asymptotic

This page as plain text: convex-polygon-diagonal-intersection-asymptotic.md

This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net^[1], current as of August 1, 2026.

1References

Packet source. MathOverflow question 212361, “Minimum distinct diagonal intersections in a convex polygon,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Question 212361 and all visible comments, checked through the Stack Exchange API on 2026-07-27.Also cited at Full question, answers, and visible comments concerning Minimum distinct diagonal intersections in a convex polygon; checked 2026-08-01.Also cited at Editorial research route recorded 2026-08-01.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Minimum distinct diagonal intersections in a convex polygon: This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.Source named by the research packet.
Bjorn Poonen and Michael Rubinstein, “The Number of Intersection Points Made by the Diagonals of a Regular Polygon,” SIAM Journal on Discrete Mathematics 11(1) (1998), 135-156. DOI 10.1137/S0895480195281246. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at main formula and asymptotic count for distinct diagonal-intersection points of the regular n-gon.Source used to assess the problem's recorded status.For Minimum distinct diagonal intersections in a convex polygon, this source supplies the regular-polygon construction and its n^4/24 leading term; it does not prove the minimum over all convex n-gons.
OEIS Foundation Inc., entry A230281, checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗reference database · reference source · checked 2026-08-01Source use: original summary.Reused material: sequence name, values a(3) through a(8), comments giving a(10) <= 157, and links to the regular-polygon sequence.Reuse basis: fair use reviewed · rights holder: The OEIS Foundation Inc. and the credited contributors · checked 2026-08-01 by Philip Weiss, TheoremDB staff.Required attribution: OEIS Foundation Inc., entry A230281, checked 2026-08-01.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at sequence name, values a(3) through a(8), comments giving a(10) <= 157, and links to the regular-polygon sequence.Source used to assess the problem's recorded status.For Minimum distinct diagonal intersections in a convex polygon, this source records the minimization sequence, verified small values, and current conjectural or upper-bound data.

This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.

[#P2882] Minimum distinct diagonal intersections in a convex polygon

1Context

2Remarks

3What counts as a solution

1Status

1Records

Computational notes

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Report a problem

1Context

2Remarks

3What counts as a solution

1Status

1Records

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance