ProofAtlas

Discrete and convex geometry

Kneser–Poulsen conjecture

Collaboration beta

If every pair of centers moves no farther apart, must the union of their balls lose volume and their common intersection gain volume, even when the radii differ?

Q≼P⇒vol(∪B(qi,ri))≤vol(∪B(pi,ri)),vol(∩B(qi,ri))≥vol(∩B(pi,ri))
Known results and sources
Two configurations of overlapping disks are connected by inward motion lines, while the union and common overlap remain visibly unresolved.
A pairwise contraction moves disk centers closer while preserving their radii; the image leaves the resulting union and intersection volume comparison explicitly open.

Research problem

Exact mathematical statement

Let P=(p₁,…,p_N) and Q=(q₁,…,q_N) be labeled configurations in Euclidean space ℝ^d with

|qi-qj|≤|pi-pj|for alli<j.|q_i-q_j|\le |p_i-p_j|\quad\text{for all }i<j.

For arbitrary positive radii r₁,…,r_N, the Kneser–Poulsen conjecture predicts

vold(⋃iB(qi,ri))≤vold(⋃iB(pi,ri))\operatorname{vol}_d\left(\bigcup_i B(q_i,r_i)\right)\le \operatorname{vol}_d\left(\bigcup_i B(p_i,r_i)\right)

and

vold(⋂iB(qi,ri))≥vold(⋂iB(pi,ri)).\operatorname{vol}_d\left(\bigcap_i B(q_i,r_i)\right)\ge \operatorname{vol}_d\left(\bigcap_i B(p_i,r_i)\right).

The retained source says that the full arbitrary-radius problem remains open and claims no full proof.

Problem infographic

Problem at a glance

A landscape plate shows a labeled configuration of unequal-radius balls contracting pairwise, alongside open union-volume and intersection-volume inequalities.
Pairwise center contraction is defined independently of the radii; the conjecture predicts a smaller union and larger intersection, and both arbitrary-radius statements are marked open.

Current mathematical picture

Where work on Kneser–Poulsen conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureIndividual adverse-residual sign forcing

The source constructs a configuration with all five adverse residuals simultaneously present while reserve terms compensate and the total gap is zero. A capacity or second-variation invariant may prove nonnegativity of the total gap even when every individual residual is adverse.

Route status · Narrowed route
Main reductionCurrent reduction

The source organizes the intersection problem by active-center shells, first proving proper-subfamily and radial-localization cases, then isolating the unresolved five-active-center benchmark and a global extension step.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve a capacity or second-variation inequality on the complete five-active-center benchmark interval.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

Kneser–Poulsen conjecture in numbers

1.6kretained lines of mathematical investigation1,580 in the current working snapshot
Argument development
1,357 · 86%
Explored or eliminated routes
58 · 4%
Computational analysis
23 · 1%
Open obligations
64 · 4%
Definitions and setup
78 · 5%
7selected mapped statements1routes investigated3open questions3contribution-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.

Status checkedAug 14, 2026
Source recordedAug 14, 2026

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

11 selected steps

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

11 selected steps

Scroll horizontally to explore the route

Working route overview for Kneser–Poulsen conjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Do pairwise contractions move union and intersection volumes in opposite monotone directions? — Depends on missing premiseDo pairwise contractionsmove union and intersectionvolumes…Current reduction — Depends on missing premiseCurrent reductionAll-residual tangent simplex — Depends on missing premiseAll-residual tangent simplexClosing target — Depends on missing premiseClosing targetProper active-family shell gaps — Depends on missing premiseProper active-family shellgapsRadial localization regime — Depends on missing premiseRadial localization regimeWeighted global intersection class — Depends on missing premiseWeighted global intersectionclassIndividual adverse-residual sign forcing — stoppedIndividual adverse-residualsign forcingProve a capacity or second-variation inequality on the complete five-active-center benchmark interval. — OpenProve a capacity orsecond-variation inequalityon…Show that the benchmark inequality is stable under degeneracies, active-set transitions, and arbitrary positive radii. — OpenShow that the benchmarkinequality is stable underdegeneracies,…Extend the controlled benchmark and special subclasses to arbitrary numbers of centers and ambient dimensions. — OpenExtend the controlledbenchmark and specialsubclasses…
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.

Explored alternatives

Other routes

1 recorded
Narrowed routeIndividual adverse-residual sign forcing

The source constructs a configuration with all five adverse residuals simultaneously present while reserve terms compensate and the total gap is zero. A capacity or second-variation invariant may prove nonnegativity of the total gap even when every individual residual is adverse.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Prove a capacity or second-variation inequality on the complete five-active-center benchmark interval.Suggested move: Differentiate the shell functional on the tangent-simplex family and identify a coercive total invariant including reserve terms.
Ready to work on
02
Show that the benchmark inequality is stable under degeneracies, active-set transitions, and arbitrary positive radii.Suggested move: Audit endpoint continuity and equality cases as centers or radii cross active-shell thresholds.
Ready to work on
03
Extend the controlled benchmark and special subclasses to arbitrary numbers of centers and ambient dimensions.Suggested move: Build an induction or decomposition that preserves total shell compensation without assuming equal radii.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The full arbitrary-radius union and intersection conjectures remain open. The contemporary specialist survey separates those universal statements from the proved planar theorem and other equal-radius or restricted-configuration cases.

[1]
External progress

What the literature has established

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

  1. Peer reviewedAishwarya and Li proved an information-theoretic Kneser–Poulsen counterpart for entropy under Gaussian heat flow. This analogue does not prove the geometric volume conjecture.[3]
  2. Authoritative summaryThe survey Selected topics from the theory of intersections of balls synthesizes Kneser–Poulsen-type problems and retains the general arbitrary-radius conjecture as open.[1]
  3. Peer reviewedBezdek and Connelly proved the Kneser–Poulsen conjecture in the Euclidean plane, a dimension-specific theorem rather than the full all-dimensional statement.[2]
3 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusKneser–Poulsen conjecture
Solved special caseplanar Kneser–Poulsen theorem

The planar Kneser–Poulsen theorem proves the contraction inequalities in dimension two.

[2]
Related problementropic Kneser–Poulsen phenomenon

The entropy theorem proves an information-theoretic counterpart for contractions and Gaussian smoothing, not the geometric union or intersection volume statement.

[3]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA statement-aligned formal definition of labeled Euclidean contractions with independently varying positive radii.
  • Formalization targetFormal Lebesgue volume of finite unions and intersections of balls, including boundary-null and continuity results.
  • Formalization targetChecked proofs of the dimension-raising, continuous-motion, or shell-integration lemmas needed by a selected route.

Research-record corrections

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.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements3 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction1 of 71
  • lemma4 of 74
  • counterexample1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementDo pairwise contractions move union and intersection volumes in opposite monotone directions?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementProper active-family shell gapsintermediate
  • retained route statementRadial localization regimeintermediate
  • retained route statementAll-residual tangent simplexintermediate
  • retained route statementWeighted global intersection classintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureIndividual adverse-residual sign forcingreported failure
  • Research targetProve a capacity or second-variation inequality on the complete five-active-center benchmark interval.open
  • Research targetShow that the benchmark inequality is stable under degeneracies, active-set transitions, and arbitrary positive radii.open
  • Research targetExtend the controlled benchmark and special subclasses to arbitrary numbers of centers and ambient dimensions.open
  • Research targetArbitrary-radius volume monotonicitysuperseded
  • Research targetFive-center capacity frontiersuperseded
  • Narrowed routeIndividual adverse-residual sign forcingThe source constructs a configuration with all five adverse residuals simultaneously present while reserve terms compensate and the total gap is zero. A capacity or second-variation invariant may prove nonnegativity of the total gap even when every individual residual is adverse.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve a capacity or second-variation inequality on the complete five-active-center benchmark interval.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointProve a capacity or second-variation inequality on the complete five-active-center benchmark interval.

Kneser–Poulsen conjecture · ready to start

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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

If every pair of centers moves no farther apart, must the union of their balls lose volume and their common intersection gain volume, even when the radii differ?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
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Sources and references3 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Selected topics from the theory of intersections of ballssurvey or monograph · Károly Bezdek, Zsolt Lángi, Márton Naszódi · Discrete Applied Mathematics · 2025-11 · ARXIV 2411.10302 · DOI 10.1016/j.dam.2025.11.040 · accessed Aug 14, 2026
  2. 2
    Pushing disks apart — the Kneser–Poulsen conjecture in the planepeer reviewed result · Károly Bezdek, Robert Connelly · Journal für die reine und angewandte Mathematik · 2002 · ARXIV math/0108098 · DOI 10.1515/crll.2002.101 · accessed Aug 14, 2026
  3. 3
    The Kneser–Poulsen Phenomena for Entropypeer reviewed result · Gautam Aishwarya, Dongbin Li · International Mathematics Research Notices · 2025-06-11 · DOI 10.1093/imrn/rnaf140 · accessed Aug 14, 2026

Important qualifications

  • This record concerns both union decrease and intersection increase under arbitrary-radius Euclidean contractions; equal-radius or planar results are not generalized.
  • The current survey is used as an authoritative specialist summary while original theorem records support the planar special case and an entropy analogue.
  • No packet URL or attachment was fetched, executed, rendered, or treated as external evidence.
  • No literature computation or geometric case analysis was independently reproduced.

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