ProofAtlas

Optimal escape paths and convex geometry

Bellman’s lost-in-a-forest problem

Collaboration beta

For a known convex forest but an unknown starting point and heading, find a shortest path that guarantees reaching its boundary. The source reports exact solutions for restricted families and local contact systems, while the general sharp construction remains open.

∀K⊂ℝ2compact convex, intK≠∅:construct an optimalγKand a sharp certificate forβ(K)
Known results and sources
Illustration: A convex forest contains translated and rotated copies of one short path, with one copy reaching the boundary.
Illustration: The forest shape is known, but the starting point and heading are unknown.

Research problem

Exact mathematical statement

For every prescribed compact convex K⊂ℝ2K\subset\mathbb R^2 with nonempty interior, determine an optimal deterministic rectifiable escape path by a general sharp construction and matching certificate. The traveler knows KK, but the starting point and heading are unknown and no observation before escape reveals the placement. Translation and orientation-preserving rotation are adversarial; first boundary contact counts as escape. Writing Γ\Gamma for the trace of a finite path prefix, the quantity to determine is

β(K)=inf{length(γ):∄t∈ℝ2, R∈SO(2) with t+RΓ⊂intK}.\beta(K)=\inf\left\{\operatorname{length}(\gamma): \nexists\,t\in\mathbb R^2,\ R\in SO(2)\ \text{with}\ t+R\Gamma\subset\operatorname{int}K\right\}.

This formula specifies the optimization problem; restating the infimum does not supply the requested construction or lower certificate. Curves may self-intersect, backtrack, pause, or have infinitely many contacts, and length includes all traversal. Reflection of an asymmetric forest is not an allowed placement.

Problem infographic

Problem at a glance

Illustration: Three translated or rotated copies of one path appear in a convex forest, beside the unknown-start rules and the shortest guaranteed-escape question.
Illustration: Find the shortest rectifiable path that cannot fit wholly inside the forest under any allowed translation or rotation. Boundary contact counts as escape.

Current mathematical picture

Where work on Bellman’s lost-in-a-forest problem stands

Recent proof claim under review

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

Useful failureRestricting polygonal dual atoms to junction anchors

K07's exact square example has width witnesses √2|cos θ| and √2|sin θ| and a sharp total dual mass √2. Atoms in the union of junction anchors and tight ties remain admissible; excluding stationary tie accumulation is still an open route.

Route status · Narrowed route
Main reductionQualified fixed-witness-graph dual

Under full-dimensional feasible-template hypotheses, K06 supplies a convex dual on the fixed maximizing witness graph. Its general-convex construction includes continuous degeneration of witness triangles to antipodal segments, but does not compare other graphs or establish finite-witness regularity for varying directions.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeControl stationary tight-tie accumulation or replace a positive-density tight set without assuming it contains intervals.Task status · Ready to work on

Work mapped so far

Bellman’s lost-in-a-forest problem in numbers

1.2kretained lines of mathematical investigation1,191 in the current working snapshot
Argument development
785 · 66%
Explored or eliminated routes
31 · 3%
Computational analysis
123 · 10%
Open obligations
139 · 12%
Definitions and setup
113 · 9%
9selected mapped statements2routes 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 checkedSep 7, 2026
Source recordedSep 7, 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 Bellman’s lost-in-a-forest problemA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Find the shortest guaranteed escape path for each compact convex forest. — Depends on missing premiseFind the shortest guaranteedescape path for each compactconvex…Current reduction — Depends on missing premiseCurrent reductionQualified fixed-witness-graph dual — Depends on missing premiseQualifiedfixed-witness-graph dualClosing target — Depends on missing premiseClosing targetFinite equilateral template realization — Depends on missing premiseFinite equilateral templaterealizationOne positive tight sum admits local selection — Depends on missing premiseOne positive tight sumadmits local selectionRestricted curved triangle family — Depends on missing premiseRestricted curved trianglefamilyTight ties remain possible atom sites — Depends on missing premiseTight ties remain possibleatom sitesTriangle optimal-pair regularity — Depends on missing premiseTriangle optimal-pairregularityRestricting polygonal dual atoms to junction anchors — stoppedRestricting polygonal dualatoms to junction anchorsShrinking total curvature before weighted allocation — stoppedShrinking total curvaturebefore weighted allocationControl stationary tight-tie accumulation or replace a positive-density tight set without assuming it contains intervals. — OpenControl stationary tight-tieaccumulation or replace apositive-density…Extend local selection to overlapping tight rows and several-changing-trace residual inequalities while retaining all affected constraints. — OpenExtend local selection tooverlapping tight rows andseveral-changing-trace…Compare unknown normal angles, temporal words, and maximizing witness graphs and obtain a lower certificate for arbitrary rectifiable curves. — OpenCompare unknown normalangles, temporal words, andmaximizing…
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 routeRestricting polygonal dual atoms to junction anchors

K07's exact square example has width witnesses √2|cos θ| and √2|sin θ| and a sharp total dual mass √2. Atoms in the union of junction anchors and tight ties remain admissible; excluding stationary tie accumulation is still an open route.

Route status · Narrowed route
Narrowed routeShrinking total curvature before weighted allocation

With component costs (4,1) and Ψ(z)=z², the source obtains optimized values 5/2 and 2 for the two specified totals. Cumulative allocation remains exact for a fixed total and specified compatible convex lower envelopes; overlapping rows and untightened negative-radius parameters require additional work.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Control stationary tight-tie accumulation or replace a positive-density tight set without assuming it contains intervals.Suggested move: Suggested next step: Analyze the one-sided stationary tie conditions and use the source's localization results to identify genuinely coupled bad regions before claiming a finite interval cover.
Ready to work on
02
Extend local selection to overlapping tight rows and several-changing-trace residual inequalities while retaining all affected constraints.Suggested move: Suggested next step: Formulate a compatibility problem for cumulative shadow allocations that preserves mass, first moments, endpoint jets, every heading, and the temporal word; keep separator parameters explicit.
Ready to work on
03
Compare unknown normal angles, temporal words, and maximizing witness graphs and obtain a lower certificate for arbitrary rectifiable curves.Suggested move: Suggested next step: Choose a concrete branch and derive an all-heading feasibility certificate and chronology-sensitive continuum lower bound, then explicitly discharge its branch restriction.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026
Current statusRecent proof claim under review

The general problem of shortest guaranteed escape for an arbitrary specified convex forest remains open in this record. Recent preprints claim a general framework and an exact solution for one triangle; no independent adjudication of those claims was established in this collection.

[1][2][3]
External progress

What the literature has established

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

  1. PreprintDeng's revised preprint claims equivalence and convergence results for a general covering formulation; its comments advertise Lean proofs of selected framework theorems.[3]
  2. PreprintTemerev and Doria claim an exact seven-piece escape path for the golden gnomon, the isosceles triangle with unit equal sides and apex angle 108°. This is a specific forest.[4]
  3. Peer reviewedGeorgiou solves an average disk-inspection variant using Fermat's principle. Its average objective and starting conditions differ from the general worst-case escape question.[5]
  4. Peer reviewedFinch and Wetzel surveyed guaranteed escape curves for known convex forests, including solved shapes and the relation to covering problems.[1]
5 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusBellman’s lost-in-a-forest problem
Related problemMoser's worm covering problem

Escape-curve and covering questions are connected in the survey; they should not be treated as identical optimization statements.

[1]
Related problemAverage disk inspection

The searcher starts at the disk's center and minimizes average inspection time, rather than the worst escape length over unknown starts and headings in a specified convex forest.

[5]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • formal proof · partial resource linkedReported Lean proofs of covering-framework theorems

    The preprint advertises Lean 4 proofs of Theorems 1–7. Their compilation, hypotheses and alignment with the full escape problem were not checked here.

    [3]
  • formal proof · partial resource linkedReported golden-gnomon finite certificate formalization

    The authors report Lean verification of finite algebraic certificates and a discrete ledger. This does not establish an independently checked proof of the complete geometric argument.

    [4]
  • computation · source linked; not reproduced by ProofAtlasGeneral covering and discretization study

    Deng describes numerical formulations using covering and traveling-salesman ideas. The paper's reported computations were not reproduced here.

    [2]

Detailed research inventory

Claims, milestones, and routes in the current map

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

7 standing statements2 proposed statements3 open questions2 narrowed routes
Statements by mathematical role9 selected mapped statements
  • theorem candidate1 of 91
  • reduction2 of 92
  • lemma4 of 94
  • special case2 of 92
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included
  • retained route statementFind the shortest guaranteed escape path for each compact convex forest.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementRestricted curved triangle familyintermediate
  • retained route statementTriangle optimal-pair regularityintermediate
  • retained route statementFinite equilateral template realizationintermediate
  • retained route statementQualified fixed-witness-graph dualintermediate
  • retained route statementTight ties remain possible atom sitesintermediate
  • retained route statementOne positive tight sum admits local selectionintermediate
  • 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
  • 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 failureRestricting polygonal dual atoms to junction anchorsreported failure
  • Useful failureShrinking total curvature before weighted allocationreported failure
  • Research targetControl stationary tight-tie accumulation or replace a positive-density tight set without assuming it contains intervals.open
  • Research targetExtend local selection to overlapping tight rows and several-changing-trace residual inequalities while retaining all affected constraints.open
  • Research targetCompare unknown normal angles, temporal words, and maximizing witness graphs and obtain a lower certificate for arbitrary rectifiable curves.open
  • Narrowed routeRestricting polygonal dual atoms to junction anchorsK07's exact square example has width witnesses √2|cos θ| and √2|sin θ| and a sharp total dual mass √2. Atoms in the union of junction anchors and tight ties remain admissible; excluding stationary tie accumulation is still an open route.
  • Narrowed routeShrinking total curvature before weighted allocationWith component costs (4,1) and Ψ(z)=z², the source obtains optimized values 5/2 and 2 for the two specified totals. Cumulative allocation remains exact for a fixed total and specified compatible convex lower envelopes; overlapping rows and untightened negative-radius parameters require additional work.
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 bridgeControl stationary tight-tie accumulation or replace a positive-density tight set without assuming it contains intervals.

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.

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Prepared starting pointControl stationary tight-tie accumulation or replace a positive-density tight set without assuming it contains intervals.

Bellman’s lost-in-a-forest problem · ready to start

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Research contextPrepared context for any AI agent

For a known convex forest but an unknown starting point and heading, find a shortest path that guarantees reaching its boundary. The source reports exact solutions for restricted families and local contact systems, while the general sharp construction remains open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references5 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Lost in a Forestsurvey or monograph · Steven R. Finch, John E. Wetzel · American Mathematical Monthly 111(8), 645–654 · 2004 · accessed Sep 7, 2026
  2. 2
    A General Solution to Bellman's Lost-in-a-forest Problempreprint · Zhipeng Deng · arXiv · 2024-12-14 · ARXIV 2412.10686 · accessed Sep 7, 2026
  3. 3
    Proof and More Variations of Bellman's Lost-in-a-forest Problempreprint · Zhipeng Deng · arXiv; version 3 dated 2026-07-29 · 2026-06-12 · ARXIV 2606.13987 · accessed Sep 7, 2026
  4. 4
    The exact solution of Bellman's lost-in-a-forest problem for the golden gnomonpreprint · Alexander Temerev, Alessio Doria · arXiv · 2026-07-27 · ARXIV 2607.24483 · accessed Sep 7, 2026
  5. 5
    Optimal Average Disk-Inspection via Fermat’s Principlepeer reviewed result · Konstantinos Georgiou · STACS 2026, LIPIcs 364, Article 44 · 2026 · accessed Sep 7, 2026

Important qualifications

  • This is a scoped administrative literature search, not an exhaustive survey or an independent mathematical review.
  • Submitted packets are not external evidence. Their novel claims, corrections and proof obligations retain their separate source-reported posture.
  • No external code, proof development, data file or certificate was executed or independently reproduced during collection. Public links confer no acceptance or redistribution authority.
  • An empty formalization or recognition list means no suitable item was established in this search; it does not establish nonexistence. Unknown origin dates and original attributions are left unresolved.
  • The Finch–Wetzel MAA PDF did not open directly. Its indexed primary-source text and the official MAA 2005 prize-book description supplied the limited survey context; the full paper was not inspected.
  • Recent preprint proof and Lean claims are recorded as author reports. Neither a preprint title nor code availability establishes a solution for every forest.

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