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Curves of constant width and Lebesgue's covering problem

Ujjwal Mishra

arXiv:2608.30538v2cs.CGmath.MG

TL;DR

The paper addresses the unknown minimum area of a convex universal cover for all planar sets of diameter one. It replaces classical polygonal test sets with constant-width bodies, combines an analytic erosion reduction with exhaustive certified subdivision, and proves the lower bound 0.8344. The method’s scope remains limited by the cost of exhaustive placement-space subdivision and the gap to the best constructive upper bound.

  • Problem

    The minimum area of a convex universal cover for all planar sets of diameter one remains unknown, despite lower bounds from classical polygonal test sets.

  • Method

    The paper uses the disc, Reuleaux triangle, and Reuleaux pentagon, erodes placements within each box to fixed inner bodies, and exhaustively subdivides the resulting five-dimensional space with independent certificate verification.

  • Results

    0.8344 is proved as a lower bound for the area of every convex universal cover.

  • Takeaways & Limitations

    Changing the test sets, rather than enlarging the computation for the classical family, yields the improved lower bound.

  • Takeaways & Limitations

    Exhaustive subdivision remains the dimensional bottleneck, while the interval [0.8344, 0.8440935944] remains unresolved.

Abstract

from arXiv · show

A universal cover is a convex set in the plane that contains a congruent copy of every planar set of diameter one. Lebesgue asked in 1914 for one of least area, and the value is not known. We prove that every convex universal cover has area at least 0.8344, improving on 0.832, published in 2005, and 0.833, in a 2026 preprint, both of which come from a disc together with an equilateral triangle and a regular pentagon. Our test sets are instead curves of constant width: the disc, the Reuleaux triangle and the Reuleaux pentagon. Each contains the regular polygon it is built on, so the family is strictly larger at the same number of bodies and the same number of placement parameters, and we show that the classical configuration admits an arrangement whose hull has area below 0.8336, so no bound drawn from those three sets by this argument reaches ours. Curves of constant width were proposed for this role, and explored numerically, by Gibbs in 2014; what is added here is a proof. It consists of an analytic reduction followed by one finite computation. The reduction bounds the hull area from below over an entire box of placements at once, by eroding each Reuleaux polygon to a fixed set contained in every placement that box allows. The computation is an exhaustive subdivision of the resulting five-dimensional space, recorded as a certificate of 486,799,600 nodes and checked by a verifier independent of the search, with a rigorous bound on its floating point error some thousands of times smaller than the margin the verification attains.

1 Introduction

Lebesgue’s open universal-cover problem is advanced by replacing classical polygonal test sets with larger constant-width bodies and proving the resulting lower bound through analytic reduction and certified exhaustive computation.

  • Problem and result: A universal cover must contain a congruent copy of every planar set of diameter one, and the least possible area remains unknown.Convexity is part of the open problem, distinguishing it from the unrestricted variant.
  • Problem and result: 0.8344 is the new lower bound for the area of every convex universal cover, improving the previously refereed bound of 0.832 and a 2026 preprint’s 0.833.The earlier bounds use a disc, equilateral triangle, and regular pentagon.
  • Proof strategy: The analytic reduction bounds hull area simultaneously over each placement box by eroding Reuleaux polygons to fixed bodies contained in every allowed placement.This converts infinitely many placements in a box into a finite-searchable lower-bound problem.
  • Test sets: The proof uses the disc, Reuleaux triangle, and Reuleaux pentagon, which enlarge the classical polygonal test sets without increasing the number of bodies, placement parameters, or search structure.Each Reuleaux polygon contains the regular polygon on which it is built.
  • Certified computation: The exhaustive five-dimensional subdivision is recorded as a certificate of 486,799,600 nodes and checked by an independent verifier.The computation avoids interval arithmetic and bounds its floating-point error by 1.72 × 10^-9.
  • Classical comparison: The classical disc–triangle–pentagon family has an arrangement with hull area below 0.8336, so this argument cannot obtain the new bound from those sets.An explicit outer-polygon calculation gives area at most 0.833597897.

2 Preliminaries

The paper reduces universal-cover lower bounds to minimum hull areas of finitely many test sets, then replaces regular polygons with larger Reuleaux polygons of the same diameter without increasing placement complexity.

  • Test-set lower bounds: A universal cover must contain congruent copies of chosen diameter-one sets, hence also the convex hull of those copies.Minimizing that hull area over all rigid motions gives a lower bound for every universal cover.
  • Test-set lower bounds: The test-set bound M controls all tuples of placements simultaneously, whereas an upper bound requires only one exhibited arrangement.The paper uses the hard direction for its lower-bound computation.
  • Constant-width test sets: Bodies of constant width one have diameter one, and every diameter-one set is contained in one, making constant-width bodies maximal admissible test sets.The proof itself requires only the diameter property, while the containment result motivates the broader family.
  • Reuleaux polygons: The Reuleaux n-gon Bn is formed by intersecting unit discs centered at the vertices of a regular n-gon of diameter one.For n=3 and n=5, these are the rounded bodies B3 and B5 used with the disc.
  • Reuleaux polygons: Each Bn contains its underlying regular polygon Pn, so replacing Pn by Bn enlarges every relevant hull without changing the number of test sets or placement parameters.This containment is the paper’s central geometric improvement over the classical polygonal family.
  • Support-function machinery: Hull support functions are computed as pointwise maxima, with translations and rotations entering through shifted support functions and directional translation terms.This identity supplies the geometric representation used for later area calculations.

3 The classical family and its ceiling

The classical disc–polygon family has a certified area ceiling below the new target, while one arrangement of the Reuleaux family remains below 0.8344 and therefore cannot itself establish the theorem.

  • Classical-family ceiling: 0.8336 is an upper bound for M(D, P3, P5), so this classical test-set family cannot yield a lower bound of 0.8344.The published 0.832 and preprint 0.833 bounds are therefore close to, but below, the family’s ceiling.
  • Classical-family ceiling: 0.833597897 is the outer-polygon area obtained for one explicit arrangement of D, P3, and P5 using 1,000,000 equally spaced support directions.Intersecting finitely many supporting half-planes gives a valid outer bound on the hull area.
  • Classical-family ceiling: The validity of the classical upper bound does not depend on how the arrangement was found, because any single arrangement competes in the defining infimum.The reported arrangement was located numerically but checked directly.
  • Reuleaux-family comparison: M(D, B3, B5) ≤0.834781191 for the authors’ Reuleaux test sets, recording a hull arrangement still below the theorem’s 0.8344 lower-bound target.This value is reported as reserve for the family rather than as an ingredient needed for Theorem 1.1.

4 Reduction to a finite computation

The paper reduces the lower-bound problem to minimizing hull areas over a normalized, bounded five-dimensional placement domain. An erosion estimate then supplies a lower bound valid for every placement in each box.

  • Gauge fixing: Symmetry reduces the three rigid motions to five placement parameters: B3 translation, and B5 rotation and translation.The disc is centered at the origin and B3 has fixed orientation; B5 rotates through [0, 2π/5).
  • Confining the placements: A distant test body forces the hull area above the target, restricting both translation vectors to norm at most 0.70.If A < 0.8344, Corollary 4.5 gives |t3| ≤ 0.70 and |t5| ≤ 0.70, yielding a compact search box.
  • The erosion estimate: For each placement box, eroding the defining discs of a Reuleaux polygon produces a fixed subset contained in every allowed placement.If each corner moves by at most δ, discs of radius 1 −δ about the box-centre corners lie inside the corresponding discs at every placement.
  • The erosion estimate: Corner displacement is bounded by translation half-widths plus rotational motion, with the latter bounded by 2R5 sin(hρ/2) for B5.For B3 the bound is sqrt(hx^2 + hy^2); for B5 it adds the rotation chord term.
  • The bound on a box: The resulting eroded bodies and the disc yield a single lower bound on hull area that applies throughout each box, not merely to sampled placements.When an eroded set is empty, the search falls back on the disc-and-point estimate.

5 The certified computation

The finite computation exhaustively subdivides the placement domain, using box-wise lower bounds and a certificate independently checked from finite witness sets. Verified leaf bounds establish the target threshold with ample arithmetic margin.

  • Subdivision: The subdivision covers the domain by bisecting boxes until each stopping box has a lower bound β ≥ 0.8344.Stopped boxes cover the initial domain, and Proposition 5.1 combines their bounds with the exterior estimate.
  • Certificate: The deterministic bisection is encoded as a binary-tree certificate, so a verifier can regenerate every box from the header and traversal bits.No boxes need to be stored individually; the record uses one bit per node before indexing and padding.
  • Certificate: 486,799,600 nodes and 245,496,952 leaves were produced for τ = 0.8344, with no subtree reaching the incompleteness guard.The certificate occupied 81.3 megabytes on disk and was generated in 591.1 minutes on eight cores.
  • Verification: The verifier replaces the disc by an inscribed polygon and the eroded bodies by tested finite witness points, then evaluates their convex-hull area.Membership tests ensure the witness collection is an inner approximation, while an independent monotone-chain implementation checks sampled leaves.
  • Verification: The fixed-body erosion method reaches the same certified lower-bound goal as parameterized inner polygons without approximating the Reuleaux bodies.The witness points remain fixed across each placement box rather than varying as functions of placement parameters.
  • What the verification establishes: Theorem 1.1 follows because every stopped box verifies α(W) ≥ 0.8344, and the arithmetic correction remains far below the smallest observed margin.The smallest margin was 1.2409 × 10^-5, versus η = 1.72 × 10^-9.

6 Scope of the method

The method’s reach is limited by subdivision cost: tightening the certified margin becomes increasingly expensive, and adding a fourth body raises the dimension sharply. The present five-dimensional family retains a small unclaimed gap, but extracting it is not feasible at current measured costs.

  • 6.1 Cost against margin: The eight-dimensional family is substantially more expensive than the five-dimensional family at every shared margin range.Its fitted power-law exponent is p = 5.50, compared with p = 2.66 for the five-dimensional range.
  • 6.1 Cost against margin: The five-dimensional cost exponent varies across scales, rising from 3.50 to 3.60 before falling to 1.94 at the tightest margins.This indicates that the minimizing neighbourhood is not self-similar, so a single whole-range power law misstates costs at different margins.
  • 6.2 What remains in the present family: The remaining gap to the proved family ceiling is about 3.8×10^-4, but certifying 0.8346 would require roughly 1.9×10^9 boxes and near 27 hours.The estimate uses the local exponent 1.94 from the two tightest margins, and the computation was not carried out.
  • 6.3 A fourth test body: Adding the Reuleaux heptagon raises the parameter count from five to eight and makes certifying 0.8344 require about 8.6×10^11 boxes, or near 880 days.The attainable value is numerically located at 0.836494901, and the method stops at this eight-dimensional case in practical terms.
  • 6.3 A fourth test body: The obstruction is computational cost rather than a failure of principle: the analytic statements and subdivision would still apply to larger finite families.Reaching the value Gibbs estimates for five bodies would require eleven parameters, further increasing the cost.

7 Related problems

The paper separates its analytic reduction from one exhaustive computation and identifies certification cost as the central barrier to extending the approach. A non-exhaustive lower-bound method would remove that dimensional obstacle.

  • Proof architecture: The computation enters only once, as a subdivision of one compact box, and establishes the hypothesis of Proposition 5.1.Sections 2–4 are analytic and contain no computation.
  • Method ceilings: The method’s constants are sharp for the chosen test sets: 0.8336 for the classical family and 0.83479 for the present family.Improving either ceiling requires enlarging the family rather than merely improving certification.
  • Computational barrier: For a fourth test body, every analytic statement and the subdivision still apply, but the box count becomes the obstacle to progress.This is the same barrier previously recorded for the polygonal family.
  • Generality: Lemma 4.7 extends beyond Reuleaux polygons to admissible constant-width test sets and can express optimization over irregular Reuleaux-polygon shapes.The paper does not attempt the resulting optimization cost.
  • Future direction: A dual or variational lower-bound argument could avoid exhausting placement space and remove the dimensional barrier.The paper locates the unresolved gap at [0.8344, 0.8440935944].
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