Convex geometry · lattice packing · optimal control · calculus of variations

Reinhardt Conjecture

Collaboration beta

Among centrally symmetric convex disks, is the smoothed octagon the unique affine shape with the lowest optimal lattice-packing density? The source reports substantial finite-dimensional reductions but no full proof.

δ (K) = \frac{area (K)}{area (H_{K})} \geq δ_{oct}

Known results and sources

The smoothed octagon is the conjectured extremal shape, not a proved optimizer.

Research problem

Exact mathematical statement

Let $K\subset\mathbb R^2$ be a centrally symmetric convex disk and let $H_K$ be its least-area circumscribed centrally symmetric hexagon. Define

\delta(K)=\frac{\operatorname{area}(K)}{\operatorname{area}(H_K)}.

Reinhardt’s conjecture asserts $\delta(K)\ge\delta_{\mathrm{oct}}$ , where equality occurs only for the affine class of the smoothed octagon and

\delta_{\mathrm{oct}}=\frac{8-\sqrt{32}-\log 2}{\sqrt 8-1}\approx0.902414182997.

The retained source explicitly states that the full conjecture is not proved.

Problem infographic

Problem at a glance

Explainer comparing a centrally symmetric convex disk, its least-area circumscribed hexagon, lattice packing, and the candidate smoothed octagon. — Reinhardt’s exact density question, the candidate value, and the distinction between the 2024 smoothed-polygon theorem and the still-open octagon selection.

Current mathematical picture

Where work on Reinhardt Conjecture stands

Open conjecture

The v20 source reports regenerated finite-support certificates and exact local primitive progress while preserving the full Reinhardt conjecture and its global closing branches as open.

Leading routePrimitive constant-conductance route

The source prioritizes parameter confinement, scalar index inequalities, and global target-fibre degree, with winding/scale separation as an alternative.

Route status · Active route

Useful failureEuclidean chart-gradient pairing

The source retains a high-precision admissible near-boundary counterexample and says not to revisit the route without changing the metric. Use the canonical physical tangent flow, total positivity, boundary asymptotics, or a rigorous interval cover of the compact core.

Route status · Eliminated route

Priority open bridgeP1 — primitive parameter confinement

Prove a parameter bound entering the low-scale box, or exclude its high-slope and high-scale complement.

Task status · Ready to work on

Later mathematical updatev20 records exact finite-support and local primitive progress

The v20 source reports regenerated finite-support certificates and exact local primitive theorems while preserving the global conjecture as open.

v20 source revision order; not a claim of occurrence time

Work mapped so far

Reinhardt Conjecture in numbers

12.6kretained lines of mathematical investigation12,629 in the current working snapshot

Argument development: 10,664 · 84%
Explored or eliminated routes: 259 · 2%
Computational analysis: 345 · 3%
Open obligations: 700 · 6%
Definitions and setup: 661 · 5%

9selected mapped statements4routes investigated7open questions7contribution-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 byKarl ReinhardtProposed1934Status checkedAug 14, 2026

Source recordedAug 21, 2026

Recommended next task

P1 — primitive parameter confinement

Prove a parameter bound entering the low-scale box, or exclude its high-slope and high-scale complement.

Suggested move: Prove a parameter bound entering the low-scale box, or exclude its high-slope and high-scale complement.

What would count as progress

Supply an exact source-independent argument or scoped counterexample with every imported premise identified.
Survive a separate mathematical review of statement scope and dependency closure.

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

20 selected steps

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

Scroll horizontally to explore the route

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.

Active routePrimitive constant-conductance route

The source prioritizes parameter confinement, scalar index inequalities, and global target-fibre degree, with winding/scale separation as an alternative.

Route status · Active route

Active routeLarger-support and global routes

The source separately keeps corrected-phase gluing, variable-conductance descent, boundary compactification, arbitrary-word no-backtracking, global compactness, and equality classification open.

Route status · Active route

Explored alternatives

Other routes

2 recorded

Eliminated routeEuclidean chart-gradient pairing

Route status · Eliminated route

Eliminated routeLocal reversal cost without endpoint control

The source says the local residual does not preserve the full SL2 endpoint. Construct an endpoint-preserving replacement or a global calibration.

Route status · Eliminated route

Route statements and reductions

Statements the next route can inspect and build on

Route statementv20 finite-support and primitive reduction

The source reduces the immediate constant-conductance frontier to primitive parameter confinement, two scalar index inequalities, target-fibre degree or winding separation, while retaining separate global branches.