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Discrete geometry · convex bodies · illumination and covering

Hadwiger–Boltyanski Illumination Conjecture

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Can the boundary of every full-dimensional convex body in ℝ^d be illuminated by at most 2^d directions, with equality only for parallelotopes? The conjecture is known for many special families but remains open in general.

I(K)≤2d
Known results and sources
External rays illuminate a general convex body beside a parallelotope while the bound I of K at most 2 to the d and its equality case remain open questions.
The conjecture asks for at most 2^d illuminating directions for every convex body, with equality only for parallelotopes; the general case is open.

Research problem

Exact mathematical statement

For every full-dimensional compact convex body K⊂ℝdK\subset\mathbb R^d with d≥2d\ge2, let I(K)I(K) be the least number of directions needed to illuminate every boundary point of KK. The Hadwiger–Boltyanski Illumination Conjecture asserts

I(K)≤2d,I(K)\le 2^d,

with equality if and only if KK is a parallelotope, equivalently an affine image of the dd-dimensional cube.

Problem infographic

Problem at a glance

A four-panel landscape defines a convex body, shows exterior illumination rays entering its interior, states the 2^d bound, and separates the parallelotope equality question from known special families.
Illumination requires rays through every boundary point to enter the interior; special families are known, but the general bound and equality classification remain open.

Current mathematical picture

Where work on Hadwiger–Boltyanski Illumination Conjecture stands

Open conjecture

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

Useful failureTreating finite combinatorial or homological verifier output as the full geometric theorem

The recorded limitation records counterexamples to independent single-face bounds, cyclic-polytope extremality, and pseudomanifold-only arguments. The residual finite cap analysis remains source-proposed as a local branch program, but only after exact cut closure and independent proof audit, followed by the unresolved broader profiles and global mechanisms.

Route status · Narrowed route
Main reductionEight residual cuts in one profile

Within the source's simplicial (1,1,2) profile, exactly the cuts 20 through 34 of even parity remain.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeClose the cut-20 sectors by proving κ=20 implies Λ≥9 or by isolating an exact finite survivor list with complete records.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

Hadwiger–Boltyanski Illumination Conjecture in numbers

2.8kretained lines of mathematical investigation2,779 in the current working snapshot
Argument development
2,263 · 81%
Explored or eliminated routes
116 · 4%
Computational analysis
55 · 2%
Open obligations
134 · 5%
Definitions and setup
211 · 8%
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.

Proposed byHugo Hadwiger; independently reformulated in illumination form by V. G. BoltyanskiProposed1957Status checkedAug 15, 2026
Source recordedAug 15, 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 Hadwiger–Boltyanski Illumination 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 2ᵈ directions always illuminate K, with equality only for parallelotopes? — Depends on missing premiseDo 2ᵈ directions alwaysilluminate K, with equalityonly…Current reduction — Depends on missing premiseCurrent reductionEight residual cuts in one profile — Depends on missing premiseEight residual cuts in oneprofileExact bound and equality target — Depends on missing premiseExact bound and equalitytargetClosing target — Depends on missing premiseClosing targetEquality classification remains separate — Depends on missing premiseEquality classificationremains separateNo complete proof claimed — Depends on missing premiseNo complete proof claimedTreating finite combinatorial or homological verifier output as the full geometric theorem — stoppedTreating finitecombinatorial or homologicalverifier…Close the cut-20 sectors by proving κ=20 implies Λ≥9 or by isolating an exact finite survivor list with complete records. — OpenClose the cut-20 sectors byproving κ=20 implies Λ≥9 orby…Independently audit the source-reported (1,1,2) classifications and the exact uses of mod-two homology. — OpenIndependently audit thesource-reported (1,1,2)classifications…Extend beyond the residual simplicial (1,1,2) branch without assuming that local cap closure proves the general conjecture. — OpenExtend beyond the residualsimplicial (1,1,2) branchwithout…
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 routeTreating finite combinatorial or homological verifier output as the full geometric theorem

The recorded limitation records counterexamples to independent single-face bounds, cyclic-polytope extremality, and pseudomanifold-only arguments. The residual finite cap analysis remains source-proposed as a local branch program, but only after exact cut closure and independent proof audit, followed by the unresolved broader profiles and global mechanisms.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Close the cut-20 sectors by proving κ=20 implies Λ≥9 or by isolating an exact finite survivor list with complete records.Suggested move: Treat the (0,20), (6,14), and (10,10) sectors separately and do not import the old cut-10 conclusion into the lower-heavy sector.
Ready to work on
02
Independently audit the source-reported (1,1,2) classifications and the exact uses of mod-two homology.Suggested move: Check the analytic normal forms, small-H classifications, non-two-pair cut-16 deduction, and verifier interpretation before publication treatment.
Ready to work on
03
Extend beyond the residual simplicial (1,1,2) branch without assuming that local cap closure proves the general conjecture.Suggested move: Analyze later label profiles and obtain a coarsening-stable inequality or finite refinement theorem for nonsimplicial cross-sections.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

The unrestricted illumination bound and parallelotope equality characterization remain a central long-standing open problem. Official sources record many solved subfamilies, including recent 1-unconditional cases, without upgrading the general status.

[1][2][3]
External progress

What the literature has established

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

  1. Peer reviewedA current Combinatorica article restated the classical conjecture as a central long-standing problem and catalogued many verified subfamilies.[1]
  2. PreprintSun and Vritsiou proved the conjecture for all 1-unconditional bodies in dimensions three and four and specified higher-dimensional subclasses.[3]
  3. Authoritative summaryBezdek and Khan surveyed the covering–illumination equivalence, special cases, and the conjecture's open status.[2]
  4. Historical sourceHadwiger posed the covering problem in 1957; later equivalent illumination formulations produced the modern named conjecture.[1]
3 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusHadwiger–Boltyanski Illumination Conjecture
Equivalent formulationLevi–Hadwiger covering problem

Illumination by directions or external light sources is equivalent to covering by translates of the interior; the exact conventions must remain aligned.

[1][2]
Solved special case1-unconditional bodies in R^3 and R^4

The conjecture is settled for this special family, together with specified higher-dimensional cases; arbitrary convex bodies remain outside the result.

[3]

Formalization opportunities

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

  • Formalization targetFormal definitions of convex body, boundary illumination, illumination number, and parallelotope in every finite dimension.
  • Formalization targetA checked equivalence between direction, external-light-source, and covering formulations under the chosen conventions.
  • Formalization targetA checked proof of both the 2^d upper bound and the equality-only-if classification for all full-dimensional convex bodies.

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
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
  • reduction2 of 72
  • lemma1 of 71
  • equivalence1 of 71
  • negative result2 of 72
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 2ᵈ directions always illuminate K, with equality only for parallelotopes?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact bound and equality targetintermediate
  • retained route statementNo complete proof claimedintermediate
  • retained route statementEquality classification remains separateintermediate
  • retained route statementEight residual cuts in one profileintermediate
  • 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 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
  • 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 failureTreating finite combinatorial or homological verifier output as the full geometric theoremreported failure
  • Research targetClose the cut-20 sectors by proving κ=20 implies Λ≥9 or by isolating an exact finite survivor list with complete records.open
  • Research targetIndependently audit the source-reported (1,1,2) classifications and the exact uses of mod-two homology.open
  • Research targetExtend beyond the residual simplicial (1,1,2) branch without assuming that local cap closure proves the general conjecture.open
  • Research targetAll three cut-20 sectors opensuperseded
  • Research targetBroader geometric program remainssuperseded
  • Narrowed routeTreating finite combinatorial or homological verifier output as the full geometric theoremThe recorded limitation records counterexamples to independent single-face bounds, cyclic-polytope extremality, and pseudomanifold-only arguments. The residual finite cap analysis remains source-proposed as a local branch program, but only after exact cut closure and independent proof audit, followed by the unresolved broader profiles and global mechanisms.
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 bridgeClose the cut-20 sectors by proving κ=20 implies Λ≥9 or by isolating an exact finite survivor list with complete records.

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 pointClose the cut-20 sectors by proving κ=20 implies Λ≥9 or by isolating an exact finite survivor list with complete records.

Hadwiger–Boltyanski Illumination Conjecture · ready to start

Mathematical updatesFollow this problem

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Research contextPrepared context for any AI agent

Can the boundary of every full-dimensional convex body in ℝ^d be illuminated by at most 2^d directions, with equality only for parallelotopes? The conjecture is known for many special families but remains open in general.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 cited works · next context review by Nov 15, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    The complex Illumination problempeer reviewed result · Liran Rotem, Alon Schejter, Boaz A. Slomka · Combinatorica 46 · 2026-01-07 · DOI 10.1007/s00493-025-00195-7 · accessed Aug 15, 2026
  2. 2
    The geometry of homothetic covering and illuminationsurvey or monograph · Károly Bezdek, Muhammad A. Khan · Springer Proceedings in Mathematics & Statistics 234 · 2018 · ARXIV 1602.06040 · accessed Aug 15, 2026
  3. 3
    Illuminating 1-unconditional convex bodies in R^3 and R^4, and certain cases in higher dimensionspreprint · Wen Rui Sun, Beatrice-Helen Vritsiou · arXiv; accepted by Canadian Journal of Mathematics · 2024 · ARXIV 2407.11331 · DOI 10.4153/S0008414X25101260 · accessed Aug 15, 2026

Important qualifications

  • The bounded pass checked bibliographic identity, current status, and representative official special-case sources; it was not an exhaustive bibliography or priority review.
  • The private packet and its URLs were not used as external authority. No submitted URL or attachment was fetched, executed, compiled, or rendered.
  • No formal proof of the unrestricted statement was identified in this bounded pass; absence from the pass is not proof of absence.

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