Browse
Claims
Search the graph by meaning. Each result carries its current verdict; open one to see its decomposition, provenance, and the reasoning behind the assessment.
ShowingImportancePrizesTopicKakeya conjecture
The conjecture in Euclidean geometry and harmonic analysis that Besicovitch sets — containing a unit line segment in every direction — must have full Hausdorff dimension. Claims about its truth, partial results, proofs, or which other conjectures depend on it belong here.
Any counterexample to the three-dimensional Kakeya conjecture must be a grainy set.
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Wang–Zahl methodHarmonic analysisimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Every three-dimensional Kakeya set has Hausdorff and Minkowski dimension at least 2.5.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →importance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Every Kakeya set in R^n has Hausdorff dimension at least (n+2)/2 (Wolff's 1995 bound).
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Harmonic analysisimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Wang–Zahl proof of the three-dimensional Kakeya conjecture can be adapted to prove the conjecture in higher dimensions.
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Harmonic analysisimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Extending the Wang–Zahl method to Kakeya in dimensions four and higher faces primarily technical rather than fundamental obstacles.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Harmonic analysisWang–Zahl methodimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The restriction, Bochner–Riesz, and local smoothing conjectures each imply the Kakeya conjecture.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Harmonic analysisimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The proof of the three-dimensional Kakeya conjecture is a once-in-a-century mathematical result.
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →Wang–Zahl methodHarmonic analysisimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The three-dimensional Kakeya conjecture was a central open problem in harmonic analysis for about fifty years.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Harmonic analysisimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The jump from two to three dimensions is the hardest step in resolving the Kakeya conjecture in all dimensions.
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →Harmonic analysisimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Kakeya set conjecture remains open in dimensions four and higher.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Harmonic analysisimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Every set of delta-tubes pointing in delta-separated directions satisfies the polynomial Wolff axioms in every dimension.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Harmonic analysisPolynomial Wolff axiomsConvex Wolff axiomsimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The convex Wolff axioms version of the Kakeya conjecture fails in dimensions four and above.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Harmonic analysisConvex Wolff axiomsimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The three-dimensional Kakeya conjecture has been proved: every three-dimensional Kakeya set has dimension three.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Harmonic analysisimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
A small-volume union of tubes in R^3 whose points lie in tubes pointing in three quantitatively different directions clusters into rectangular slabs (grains) at an intermediate scale.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Wang–Zahl methodHarmonic analysisimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Fourier restriction conjecture implies the Kakeya conjecture.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Harmonic analysisFourier restriction conjectureimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The mathematical community accepts the Wang–Zahl proof of the three-dimensional Kakeya conjecture as correct.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Wang–Zahl methodMathematicsHarmonic analysisimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Hong Wang and Joshua Zahl posted an arXiv preprint in February 2025 claiming a proof of the three-dimensional Kakeya set conjecture.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Harmonic analysisWang–Zahl methodimportance · settledImportance 0.12, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
There exists a planar set of arbitrarily small (zero) area through which a unit line segment can be rotated through every direction
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →importance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Contribute
If a claim here is wrong, or missing evidence, open it: every claim page carries its own entry for challenges, evidence, and corrections. If the graph is missing a claim entirely, propose it below. A proposal is reviewed on its merits; accepted claims are matched against the graph and enter it with their reasoning on record.