# P2882: Minimum distinct diagonal intersections in a convex polygon
- ID: `P2882`
- Reference: `convex-polygon-diagonal-intersection-asymptotic`
- Page: https://theoremdb.org/statements/P2882
- Record maturity: Reviewed problem with recorded work
## 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 \(n\to\infty\)?
### Remarks
- **Remark.** A diagonal joins two nonconsecutive vertices of the polygon.
- **Remark.** A strictly convex polygon has distinct vertices in convex position and every interior angle strictly smaller than pi.
### What 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.
## Status
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](#reference-2) [1](#reference-1) [3](#reference-3)
## Work
### Evidence for the current status
**Claim 1 (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.
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.
### Background and intake notes
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.
- 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.
- Recorded example: For n=4 one has f(4)=1, and a regular hexagon has 13 distinct interior diagonal-intersection points.
### Open directions
- **Route 1** (reported): 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](#reference-1)
### 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.
### Working on this
Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `convex-polygon-diagonal-intersection-asymptotic`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.
## References
1. MathOverflow question 212361, “Minimum distinct diagonal intersections in a convex polygon,” checked 2026-08-01. Question 212361 and all visible comments, checked through the Stack Exchange API on 2026-07-27. https://mathoverflow.net/questions/212361/minimal-number-of-intersections-in-a-convex-n-gon
- 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.
- forum; reference source; checked 2026-08-01
- Source use: citation_only
- 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.
2. 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. main formula and asymptotic count for distinct diagonal-intersection points of the regular n-gon https://doi.org/10.1137/S0895480195281246
- scholarly_publication; reference source; checked 2026-08-01
- Source use: citation_only
- 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.
3. OEIS Foundation Inc., entry A230281, checked 2026-08-01. sequence name, values a(3) through a(8), comments giving a(10) <= 157, and links to the regular-polygon sequence https://oeis.org/A230281
- reference_database; reference source; checked 2026-08-01
- Source use: citation_only
- 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.
- 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.