Convex geometry · geometric probability · additive energy

Hall’s Random-Triangle Conjecture

Collaboration beta

Among planar convex regions of a fixed area, does the disk give the largest chance that three independent uniform points form an acute triangle?

\forall K \subset ℝ^{2} bounded and convex, p (K) \leq p (D) = \frac{4}{π^{2}} - \frac{1}{8}

Known results and sources

Three random points in a convex region form the triangle whose chance of being acute is compared with the disk.

Research problem

Exact mathematical statement

Let $p(K)$ be the probability that three independent uniform points in a bounded planar convex region $K$ form an acute triangle, and let $D$ be a disk of the same area. Hall’s conjecture asks whether

\forall K\subset\mathbb R^2\text{ bounded and convex},\qquad p(K)\le p(D)=\frac{4}{\pi^2}-\frac18.

Problem infographic

Problem at a glance

Problem-centered plate for Hall's random-triangle conjecture. A same-area convex region K and disk D each contain three independently sampled points joined into a triangle; an acute example marks all three acute angles, while a small obtuse example marks one angle greater than 90 degrees. The plate asks whether p(K) is at most p(D), gives p(D) = 4/π² − 1/8 ≈ 0.280285, lists retained special classes, and marks the general case open. — Choose three independent uniform points in a bounded planar convex region K, and let p(K) be the probability that they form an acute triangle. Hall's conjecture asks whether, among regions of the same area, the disk D always maximizes this probability: p(K) ≤ p(D), where p(D) = 4/π² − 1/8. The recorded route treats ellipses, rectangles, triangles, and parallelograms as special classes; the general convex-region case remains open.

Current mathematical picture

Where work on Hall’s Random-Triangle Conjecture stands

Open conjecture

The retained v8 source material gives exact probability, additive-energy, chord-deficit, affine-synchronization, threshold-Bregman, obstacle, Bellman, Stieltjes, and roof-rigidity reductions; records strict special-class results for ellipses, rectangles, triangles, and parallelograms; closes the smooth and centrally symmetric homothetic-level branches; and preserves exact witnesses eliminating several tempting proof routes. Its current frontier is the threshold–obstacle clipping comparison, whose load-bearing coefficient-four identity, moderate-band comparison, roof-to-triangle stability, high-core estimate, nonsmooth passage, and fragile certificates still require the stated independent audits. The conjecture remains open, and this state is a curated overview rather than a complete inventory of the 14,136-line source.

Strongest supported footholdThreshold Bregman resource and paid band

Wrong threshold area and shape are recombined into one nonnegative divergence whose total mass is L²/2 and whose first 56.9% is paid levelwise.

Evidence posture · Reported result

Leading routeCoefficient-four obstacle regeneration

Rebuild the load-bearing coefficient-four decomposition before using the current frontier as a positive comparison.

Route status · Active route

Useful failureClassify smooth or centrally symmetric homothetic cores

This branch is closed: the covered homothetic-level bodies are ellipses, so further reclassification is not a current proof task.

Route status · Eliminated route

Main reductionAll-convex action and one-level roof rigidity

The smooth reserve is replaced by an all-convex Stieltjes action, and exact roof behavior at one level is reduced to the triangle class.

Evidence posture · Reported reduction

Completed special caseFour global affine classes retained

The current work records strict Hall comparisons for noncircular ellipses, rectangles, triangles, and parallelograms, with the triangle certificate still awaiting regeneration.

Evidence posture · Reported special case

Priority open bridgeProve the moderate-band Bellman comparison

Charge the exact moderate-level clipping source to the same-threshold Bregman/Jacobi resource while preserving synchronization, sign, homogeneity, Green, overlap, and saturation remainders.

Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Hall’s Random-Triangle Conjecture in numbers

4.9kretained lines of mathematical investigation4,935 in the current working snapshot

Argument development: 4,278 · 87%
Explored or eliminated routes: 141 · 3%
Computational analysis: 76 · 2%
Open obligations: 183 · 4%
Definitions and setup: 257 · 5%

24selected mapped statements14routes investigated12reported milestones6open questions6contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byGlen Richard HallProposed1982Status checkedAug 2, 2026

Source recordedAug 2, 2026

Recommended next task

Prove the moderate-band Bellman comparison

Charge the exact moderate-level clipping source to the same-threshold Bregman/Jacobi resource while preserving synchronization, sign, homogeneity, Green, overlap, and saturation remainders.

Suggested move: Start from the exact levelwise clipping density and adjoint identity on 0≤τ≤τ*, then compare the equilibrium drift source directly with the threshold resource.

What would count as progress

Prove an exact constant comparison on the moderate band.
Leave all favorable Bellman, Green, overlap, and saturation terms explicitly subtracted.
Use only the one global affine shear.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

26 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeCoefficient-four obstacle regeneration

Rebuild the load-bearing coefficient-four decomposition before using the current frontier as a positive comparison.

Route status · Active route

Active routeModerate-band Bellman-to-threshold comparison

Combine the paid band with the exact Bellman source, Green interpolation, overlap, saturation, and secant cancellation under one global affine shear.

Route status · Active route

Active routeRoof proximity to strict triangle margin

Turn small compact-band localized roof reserve into quantitative triangle proximity and then use the regenerated strict triangle margin.

Route status · Active route

Active routeHigh-core obstacle–clipping estimate

Use secant localization, the polar invariant, top-core truncation, the disk Hessian, and 𝒪₄ to pay the residual high-core clipping term across levels.

Route status · Active route

Explored alternatives

Other routes

10 recorded

Narrowed routeAll-convex nonsmooth passage

Retain the all-convex threshold, roof, and Stieltjes quantities while smoothing only the clipping/Bellman side with explicit endpoint and semicontinuity control.

Route status · Narrowed route

Eliminated routeClassify smooth or centrally symmetric homothetic cores

This branch is closed: the covered homothetic-level bodies are ellipses, so further reclassification is not a current proof task.

Route status · Eliminated route

Refuted routeL²-only clipping payment

The equilateral triangle defeats every universal attempt to pay Γ using only the threshold L² resource.

Route status · Refuted route

Browse 7 more explored routes

Refuted routePointwise energy-body inclusion

The square witness refutes pointwise inclusion at the disk ratio; an integrated angular energy-body comparison remains possible.

Route status · Refuted route

Useful but insufficientOne-dimensional stochastic corridor alone

The triangle–body–disk orders are informative but too weak to supply the final disk-Jacobi sign without two-dimensional geometry.

Route status · Useful but insufficient

Eliminated routeSeparate value and threshold estimates

Separate positive-part bounds discard the exact secant cancellation and are structurally incompatible with disk-sharp scaling.

Route status · Eliminated route

Useful but insufficientAbstract Fourier and moment relaxation

Integrated moments and finitely many affine constraints do not encode the spatial compatibility needed for Hall.

Route status · Useful but insufficient

Useful but insufficientSeparate cap-wise reserves

Cap support, symmetry, concavity, and affine-isoperimetric reserves are too weak without quantitative use of the nonlinear cap equation.

Route status · Useful but insufficient

Refuted routeGlobal 15/8 affine-deficit ratio

The equilateral triangle refutes the global ratio, although a central-slice version remains an open testbed.

Route status · Refuted route

Eliminated routePointwise local payment of high-core clipping

Endpoint order mismatch rules out paying the high-core cubic pointwise with quartic inertia or roof reserve; cross-level obstacle compensation is required.

Route status · Eliminated route

Route statements and reductions

Statements the next route can inspect and build on

Route statementCoefficient-four obstacle decomposition

The retained coefficient-four identity decomposes 4𝓔−δ_Q as 𝓛₂+2𝒞_D+𝒪₄ with 𝒪₄≥0, hence 𝒫_obs=𝒞_D+𝒪₄/2≥0.

Source-reported route statement · dependencies incomplete

Route statementUniversal paid threshold band

For thresholds 0≤τ≤τ*=0.5693178886…, the disk Jacobi budget pays the combined wrong-area and wrong-shape threshold divergence levelwise; the defining root still needs interval certification.

Source-reported route statement · dependencies incomplete

Route statementExact Bellman propagation with favorable remainders

The levelwise clipping Bellman identity exposes synchronization, sign, and homogeneity remainders, all nonnegative, and aligns its positive source with the exact drift measure.

Source-reported route statement · dependencies incomplete

Route statementExact secant cancellation

Signed value mismatch and threshold geometry combine into a quadratic spatial term plus an outside-disk-support obstacle; estimating their positive parts separately destroys the disk scaling.

Source-reported route statement · dependencies incomplete

Route statementOne-level roof rigidity

If the localized roof reserve vanishes at any one nonzero covariogram height, then K is a triangle up to affine equivalence.

Source-reported route statement · dependencies incomplete

Route statementStrict all-triangle Hall margin

The current work reports Hall strictly for every triangle via an affine deficit ratio at least Λ△=1.8038103379…>9/5; its Bernstein tree, sector Jacobian, and interval certificate must be regenerated.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks

Prove the moderate-band Bellman comparison

Charge the exact moderate-level clipping source to the same-threshold Bregman/Jacobi resource while preserving synchronization, sign, homogeneity, Green, overlap, and saturation remainders.

Suggested move: Start from the exact levelwise clipping density and adjoint identity on 0≤τ≤τ*, then compare the equilibrium drift source directly with the threshold resource.

Ready to work on

Control the residual high-core clipping term

Charge every residual high-core negative secant contribution to the coefficient-four core-obstacle resource using cross-level compensation rather than pointwise local inertia.

Suggested move: Combine secant localization, the polar invariant, top-core truncation, the disk Hessian, and 𝒪₄ with exact constants.

Ready to work on

Regenerate the coefficient-four obstacle identity

Independently rederive the convolution expansion, affine-frame scaling, reflection orientation, and core-potential sign behind 4𝓔−δ_Q=𝓛₂+2𝒞_D+𝒪₄ and 𝒫_obs≥0.

Suggested move: Re-expand g_K−g_D from 1_K−1_D in the canonical covariance-isotropic frame and verify every coefficient and sign before estimating Γ.

Ready to work on

Transfer roof proximity to the triangle margin

Quantify how small localized roof reserve on a compact level band forces sufficient triangle proximity to invoke the strict all-triangle clipping margin.

Suggested move: Combine one-level rigidity, affine compactness, an explicit symmetric-difference continuity modulus, and a regenerated triangle certificate.

Ready to work on

Regenerate fragile exact certificates

Independently regenerate the all-triangle Bernstein and sector certificate, the equilateral mixed integral and rational >1/4000 witness, and the τ* interval root before public reliance.

Suggested move: Reimplement each retained certificate independently and preserve code, inputs, exact intervals, and digests.

Ready to work on

Make the clipping comparison all-convex

Pass the regular-level clipping, Bellman, Gauss-angle, and obstacle formulas rigorously to nonsmooth convex bodies while retaining lower semicontinuity and endpoint control.

Suggested move: Recombine the smooth identities into all-convex Stieltjes and threshold quantities before taking polygonal or smoothing limits.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 22, 2026

Current statusOpen conjecture

The conjecture that the Euclidean ball maximizes the probability that three independent uniform points in a bounded convex domain form an acute triangle remains open. The strongest located general progress is local extremality of the disk in dimension two, weak local extremality of the ball in dimension three, an explicit exclusion for planar domains of sufficiently large isoperimetric ratio, and a finite but presently infeasible reduction of the planar case.

[1]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2020
PreprintKhan proved planar convex domains with isoperimetric ratio greater than 7688/15 have smaller acute-triangle probability than the disk and reduced the full planar conjecture to a finite, currently intractable…[1]
2020
Peer reviewedKhan proved the disk is a local maximum among bounded planar convex domains and the three-dimensional ball is a weak local maximum.[2]
1982
Historical sourceHall computed the acute-triangle probability for the n-ball and observed the ball's criticality, motivating the extremal conjecture.[3]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

12 mapped milestonesretained argument map

Browse all 12 mapped stages

stage 1Exact probability and Q–D frontier
stage 2Sharp energy endpoints and affine minimization
stage 3Positive disk-obstacle resource
stage 4One globally synchronized clipping rebate
stage 5Four global affine classes retained
stage 6Exact route eliminations
stage 7Homothetic-level branches classified
stage 8Bellman, adjoint, and saturation structure
stage 9All-convex action and one-level roof rigidity
stage 10Threshold Bregman resource and paid band
stage 11Equilateral obstruction makes the obstacle resource mandatory
stage 12Threshold–obstacle frontier and first missing bridge

Exact probability and Q–D frontierThe retained source first converts acute-triangle probability into an exact additive-energy and chord-deficit inequality.