ProofAtlas

Convex geometry

Log-Brunn–Minkowski Conjecture

Collaboration beta

The conjecture says a logarithmic interpolation of two symmetric convex bodies should have at least the geometric-mean volume. This packet turns the question into concavity of box sections and proves some source-contained cap configurations, but missing certificates and an unequal-overlap case block broader conclusions.

|K#tL|≥|K|1-t|L|t
Known results and sources
Two symmetric convex bodies interpolate into a teal middle body while a pair of unequal rust caps remains unresolved at the edge.
The conjecture compares volume under logarithmic support interpolation; selected cap geometries are understood, but the general case remains open.

Research problem

Exact mathematical statement

For origin-symmetric convex bodies K,L⊂ℝnK,L\subset\mathbb R^n and t∈[0,1]t\in[0,1], define the logarithmic Wulff combination by

K#tL=⋂ω∈Sn-1{x∈ℝn:⟨x,ω⟩≤hK(ω)1-thL(ω)t}.K\#_tL = \bigcap_{\omega\in S^{n-1}} \left\{x\in\mathbb R^n: \langle x,\omega\rangle\le h_K(\omega)^{1-t}h_L(\omega)^t\right\}.

Equivalently, K#tLK\#_tL is the largest convex body whose support function is bounded above by the pointwise geometric interpolation hK1-thLth_K^{1-t}h_L^t. The conjecture is

|K#tL|≥|K|1-t|L|t.|K\#_tL| \ge |K|^{1-t}|L|^t.

Problem infographic

Problem at a glance

A wide diagram links symmetric-body interpolation to a sliced coordinate box, then separates understood cap configurations from an unresolved unequal-depth core.
The source connects the volume inequality to box-section concavity and isolates cap families, missing finite certificates, and an unequal-depth matrix target.

Current mathematical picture

Where work on Log-Brunn–Minkowski Conjecture stands

Open conjecture

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

Useful failureConvexity of removed-volume fraction

The updated do-not-revisit list says an exact counterexample is recorded in Section 9. Retain the logarithmic Euler-Hessian correction and attack the unequal-depth invariant block directly.

Route status · Narrowed route
Main reductionCurrent reduction

The source states BOX_n is equivalent to log-Brunn–Minkowski in dimension n, modulo a standard symmetric-polytopal approximation in one direction.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeReproduce the missing finite certificate families for the reported complete (N,n,k)=(5,3,2) result.Task status · Ready to work on
Research-record correctionResearch-record correction

We 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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Log-Brunn–Minkowski Conjecture in numbers

2.2kretained lines of mathematical investigation2,237 in the current working snapshot
Argument development
1,894 · 85%
Explored or eliminated routes
31 · 1%
Computational analysis
107 · 5%
Open obligations
95 · 4%
Definitions and setup
110 · 5%
8selected mapped statements2routes investigated4open questions4contribution-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 byKároly J. Böröczky, Erwin Lutwak, Deane Yang, Gaoyong ZhangProposed2012Status 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

14 selected steps

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

14 selected steps

Scroll horizontally to explore the route

Working route overview for Log-Brunn–Minkowski ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Volume stays above the geometric mean under logarithmic support interpolation. — Depends on missing premiseVolume stays above thegeometric mean underlogarithmic…BOX equivalence — Depends on missing premiseBOX equivalenceCurrent reduction — Depends on missing premiseCurrent reductionAll-moduli certificates absent — ChallengedAll-moduli certificatesabsentClosing target — Depends on missing premiseClosing targetEqual-depth adjacent overlap — Depends on missing premiseEqual-depth adjacent overlapNoninteracting cap family — Depends on missing premiseNoninteracting cap familyNinety relevant factors — Depends on missing premiseNinety relevant factorsConvexity of removed-volume fraction — stoppedConvexity of removed-volumefractionPromoting chat-derived computation — stoppedPromoting chat-derivedcomputationReproduce the missing finite certificate families for the reported complete (N,n,k)=(5,3,2) result. — OpenReproduce the missing finitecertificate families for thereported…Prove positive definiteness of the unequal-depth adjacent-overlap scale-quotient block for dimension four. — OpenProve positive definitenessof the unequal-depthadjacent-overlap…Extend verified local chamber results to a global BOX theorem without losing event-boundary control. — OpenExtend verified localchamber results to a globalBOX…Unequal-depth block — OpenUnequal-depth block
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

2 recorded
Narrowed routeConvexity of removed-volume fraction

The updated do-not-revisit list says an exact counterexample is recorded in Section 9. Retain the logarithmic Euler-Hessian correction and attack the unequal-depth invariant block directly.

Route status · Narrowed route
Narrowed routePromoting chat-derived computation

The source states the large datasets and certificate scripts supporting items B1-B6 are absent from the supplied filesystem. Reproduce the enumerations and positivity certificates in the specified portable schema, then rerun exact audit gates.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Reproduce the missing finite certificate families for the reported complete (N,n,k)=(5,3,2) result.Suggested move: Generate immutable exact sphere, circuit, ray, cone, and coefficient records with deterministic independent verification.
Ready to work on
02
Prove positive definiteness of the unequal-depth adjacent-overlap scale-quotient block for dimension four.Suggested move: Clear positive denominators, factor the three principal minors, and seek Bernstein-positive or sum-of-squares forms.
Ready to work on
03
Unequal-depth block

For n>=4, unequal-depth adjacent overlap remains a live fixed-size matrix target.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Extend verified local chamber results to a global BOX theorem without losing event-boundary control.Suggested move: Establish a composition or leaf-removal law for cap-interaction forests and audit all combinatorial transitions.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The origin-symmetric log-Brunn–Minkowski volume inequality is open in general dimensions. The planar case and substantial local or structured cases are known, but current publisher-hosted literature continues to describe the general p=0 problem as open.

[1][3][2]
External progress

What the literature has established

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

  1. Peer reviewedA current publisher-hosted paper describes the p in [0,1) Brunn–Minkowski inequality as open at large and identifies the p=0 case as the log-Brunn–Minkowski conjecture.[3]
  2. Peer reviewedKolesnikov and Milman developed local Lp-Brunn–Minkowski inequalities for p<1, including local logarithmic results and applications to uniqueness and stability.[2]
  3. Peer reviewedBöröczky, Lutwak, Yang, and Zhang formulated the origin-symmetric inequality, proved equivalence with a logarithmic Minkowski family, and established the conjectured inequalities for plane convex bodies.[1]
3 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusLog-Brunn–Minkowski Conjecture
Solved special casePlanar origin-symmetric convex bodies

The origin-symmetric conjectured inequalities are proved for convex bodies in the plane.

[1]
Weaker or relaxed formLocal Lp-Brunn–Minkowski inequalities for p<1

Local Lp-Brunn–Minkowski inequalities establish infinitesimal or neighborhood results under stated conditions rather than the unrestricted global conjecture.

[2]

Formalization opportunities

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

  • Formalization targetA formal convex-body support-function and logarithmic interpolation library
  • Formalization targetFormal volume and symmetric-polytopal approximation infrastructure
  • Formalization targetMachine-checked statements of the planar and local theorems with exact hypotheses

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 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 removed a duplicate or outdated task or route step. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Claim connections clarified

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 statements1 challenged statements4 open questions2 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction1 of 81
  • lemma3 of 83
  • equivalence1 of 81
  • computational claim1 of 81
  • negative result1 of 81
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.23 displayed rows · 2 routes included
  • retained route statementVolume stays above the geometric mean under logarithmic support interpolation.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementBOX equivalenceintermediate
  • retained route statementNinety relevant factorsintermediate
  • retained route statementNoninteracting cap familyintermediate
  • retained route statementEqual-depth adjacent overlapintermediate
  • retained route statementAll-moduli certificates absentintermediate
  • 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 failureConvexity of removed-volume fractionreported failure
  • Useful failurePromoting chat-derived computationreported failure
  • Research targetReproduce the missing finite certificate families for the reported complete (N,n,k)=(5,3,2) result.open
  • Research targetProve positive definiteness of the unequal-depth adjacent-overlap scale-quotient block for dimension four.open
  • Research targetExtend verified local chamber results to a global BOX theorem without losing event-boundary control.open
  • Research targetUnequal-depth blockopen
  • ComputationThe current work reports an exact 272-to-90 event-factor filter and a fixed projective-point certificate, while larger all-moduli classifications lack retained post-v8 artifacts.The fixed point and 90-factor reduction are labeled packet-certified; the claimed 55-chamber all-moduli completion is conditional on reproducing absent certificates. · reported unreproduced
  • Narrowed routeConvexity of removed-volume fractionThe updated do-not-revisit list says an exact counterexample is recorded in Section 9. Retain the logarithmic Euler-Hessian correction and attack the unequal-depth invariant block directly.
  • Narrowed routePromoting chat-derived computationThe source states the large datasets and certificate scripts supporting items B1-B6 are absent from the supplied filesystem. Reproduce the enumerations and positivity certificates in the specified portable schema, then rerun exact audit gates.
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 bridgeReproduce the missing finite certificate families for the reported complete (N,n,k)=(5,3,2) result.

2 approaches have 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

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointReproduce the missing finite certificate families for the reported complete (N,n,k)=(5,3,2) result.

Log-Brunn–Minkowski Conjecture · ready to start

Mathematical updatesFollow this problem

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

The conjecture says a logarithmic interpolation of two symmetric convex bodies should have at least the geometric-mean volume. this work turns the question into concavity of box sections and proves some source-contained cap configurations, but missing certificates and an unequal-overlap case block broader conclusions.

  • 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
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

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
    The log-Brunn–Minkowski inequalityoriginal source · Károly J. Böröczky, Erwin Lutwak, Deane Yang, Gaoyong Zhang · Advances in Mathematics · 2012 · DOI 10.1016/j.aim.2012.07.015 · accessed Aug 14, 2026
  2. 2
    Local Lp-Brunn–Minkowski Inequalities for p<1survey or monograph · Alexander V. Kolesnikov, Emanuel Milman · Memoirs of the American Mathematical Society · 2022 · accessed Aug 14, 2026
  3. 3
    The Lp-Brunn–Minkowski inequalities for variational functionals with 0≤p<1peer reviewed result · Calculus of Variations and Partial Differential Equations · 2025 · DOI 10.1007/s00526-025-03090-7 · accessed Aug 14, 2026

Important qualifications

  • Scoped publisher-hosted pass focused on the origin-symmetric volume inequality.
  • Results for torsion, capacity, and other variational functionals are not transferred to convex-body volume.
  • No submitted packet URL was fetched.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image