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A 0.41275 lower bound for the three-dimensional Blaschke-Lebesgue problem
Yunjiang Jiang
TL;DR
The three-dimensional Blaschke–Lebesgue problem asks for the minimum volume among constant-width bodies and which bodies attain it. This paper combines contact-tetrahedron geometry with radial estimates to prove a universal volume lower-bound theorem, without identifying a minimizer.
Problem
The three-dimensional Blaschke–Lebesgue problem asks for the minimum volume of a constant-width body and the bodies that attain it.
Source: §1 · p. 1Method
The proof combines disjoint face, edge, and face-edge regions around a balanced contact tetrahedron with closed-form radial profiles around contacts and antipodes.
Source: §1.2 · p. 3Results
Theorem 1.1 gives a volume lower bound for every convex body of constant width, with the proof covering the full Jung interval and scaling to width w.
Source: §1.1 · p. 2, §6.5 · p. 17Takeaways & Limitations
The theorem establishes a volume bound but does not identify a minimizer or prove the Meissner conjecture.
Source: §1.1 · p. 2The estimates are not optimized, and 0.41275 is not a limit of the contact-tetrahedron or radial methods.
Source: §6.5 · p. 17
Abstract
from arXiv · showhide
We prove that every convex body $K\subset\mathbb{R}^3$ of constant width $w>0$ satisfies \[ \operatorname{Vol}(K)\ge \frac{1651}{4000}w^3=0.41275\,w^3. \] This improves the certified lower bound $0.411040473721188\ldots\,w^3$ announced by Hyra and the earlier spectral bound $4\pi w^3/33$ of Nishioka. The main new ingredient is a contact-tetrahedron estimate near equality in Jung's circumradius inequality. Four balanced contacts of a minimum circumscribed ball form a near-regular tetrahedron; constant width then forces, beyond the tetrahedron itself, spherical regions over its faces, spindle sectors along its edges, and additional face-edge sectors. These regions can be selected with pairwise disjoint interiors and quantified by the total squared-edge deficit. A sharp convexity argument relates that deficit to the circumradius, and a symmetrization argument controls the four face contributions with only quadratic loss. This gives the analytic bound $\operatorname{Vol}(K)>0.41275318$ near the Jung endpoint. For the complementary circumradius range we use Hyra's radial transport/tangent-majorant mechanism together with closed-form contact and antipodal profiles. The antipodal contribution has an elementary primitive. For the contact contribution, a second-order concavity estimate converts the angular Jacobian into an explicit rational lower density; that density is monotone in the circumradius. The whole intermediate range therefore reduces to a single endpoint calculation, giving $\operatorname{Vol}(K)>0.41275168$. For smaller circumradius the tangent-majorant inequality with its nonnegative gap discarded already gives $\operatorname{Vol}(K)>0.41308109$. The proof is entirely analytic and requires no computer-assisted certificate.
1. Introduction
The three-dimensional Blaschke–Lebesgue problem asks for the minimum volume among convex bodies of prescribed constant width and for the bodies attaining it. This introduction states a universal volume bound and previews geometric and radial arguments across three circumradius regimes.
1. Introduction
- The problem asks for the infimum m3(w), which scales as w^3, and for the bodies attaining it; the minimizer conjecture has remained open for more than ninety years.Source: §1 · p. 1, §1.1 · p. 1
1.1. Historical context.
- Using a minimum-circumsphere-contact geometric method and a finite exact-rational cap certificate, Hyra obtains the stronger volume lower bound Vol(K) > 0.411040473721188 w^3.Source: §1.1 · p. 2
1.2. The proof at a glance.
- For the complementary circumradius range, closed-form contact and antipodal profiles support a radial estimate whose rational contact density decreases with R, reducing the intermediate range to R = R0.The antipodal cap's tangent-gap contribution has an elementary rational primitive.Source: §1.2 · p. 3
- Near the Jung endpoint, balanced contacts form a near-regular tetrahedron whose faces, edges, and intervening sectors force additional volume with disjoint interiors.A symmetrization of the face contributions cancels first-order anisotropy, leaving quadratic loss.Source: §1.2 · p. 3
1.3. Exact constants and computational scope.
- The proof uses three circumradius regimes and is computer-free, with terminating decimals interpreted as exact rationals.The final assembly covers the full Jung interval.Source: §1.2 · p. 3, §1.3 · p. 3, §1.4 · p. 4
2. The near-Jung geometric estimate
The section states a near-Jung geometric estimate for a width-one body, under a circumradius condition and a bound on the total squared-edge deficit.
2. The near-Jung geometric estimate
- The section introduces a quantitative estimate to be proved in Section 3 and the following two sections.Source: §2 · p. 4
- Terminating decimals in the proof denote exact rational numbers.Source: §2 · p. 4
- Theorem 2.1 applies when R ≥ 0.609 and the six nonnegative squared-edge deficits of a balanced contact tetrahedron sum to at most 0.06.The supplied theorem statement truncates before giving the resulting bound.Source: §2 · p. 4
3. Constant width and a sharper edge-deficit estimate
The section connects constant-width geometry to a balanced set of circumsphere contacts, then controls the contact tetrahedron through its squared-edge deficits.
3. Constant width and a sharper edge-deficit estimate
- A width-one body has diameter one, and each boundary point with outer normal n has its antipode x − n in the body.The width identity and unit-distance bound establish the antipodal inclusion.Source: §3 · p. 4
- A minimum enclosing ball has balanced contacts, and Caratheodory’s theorem reduces the balancing set to at most four contacts.The contacts can be written as x_i = R u_i.Source: §3 · p. 4
- For circumradius R ≥ 0.6, the squared-edge deficit sum D is bounded by 16d + 576d^2 and is less than 1.The passage also relates the contact weights to d.Source: §3 · p. 5
- Convexity over the deficit simplex bounds the relevant vertex quantity by D^2/[16(2 + D)].At a simplex vertex, one deficit equals D and the others are zero.Source: §3 · p. 5
4. Volume of the tetrahedron and strengthened face regions
The section estimates the tetrahedron and the face regions forced by the enclosing ball, then symmetrizes and compares the face contributions with a regular-face benchmark.
4. Volume of the tetrahedron and strengthened face regions
- When D ≤ 0.06, the tetrahedron’s faces are acute, have squared circumradius at most 1/3, and lie outside an origin strictly inside the tetrahedron.Source: §4 · p. 5
4.1. Using the enclosing ball.
- The enclosing-ball argument places a circumradius-aware region over each face inside K, and these regions project orthogonally into their respective faces.The region uses W(p) = R^2 − |p|^2 and κ = R + h.Source: §4.1 · p. 6
4.2. Symmetrizing the faces.
- Symmetrizing the four face contributions cancels the linear anisotropy, leaving a quadratic loss.The face contribution is bounded using the symmetrized parameter κs.Source: §4.2 · pp. 6–7
4.3. Comparison with the regular face.
- The regular-face comparison is reduced to a final inequality verified using rational bounds.Source: §4.3 · pp. 7–8
5. Edge spindles and the previously uncounted sectors
The section adds spindle volume along tetrahedron edges and extends face patches into face-edge sectors, using projection and angle control to keep the counted regions disjoint.
5.1. Spindles and projection disjointness.
- Spindle regions along edges contribute volume β_e J(ℓ_e)/2, and unique nearest-point projections make edge and face regions disjoint except on null boundaries.Source: §5.1 · p. 8
5.2. Quantitative angle control.
- The edge integral satisfies 0 ≤ J_0 − J(ℓ_e) ≤ j_0δ_e, giving quantitative control of spindle volume through the squared-edge deficit.The derivative bound is established by interpolating squared edge lengths.Source: §5.2 · pp. 8–9
5.3. Extending the spherical face patches past their projections.
- Spherical triangles above faces lie in every unit ball containing the face vertices, and the resulting face-edge flanks can be added without overlapping face regions or spindles.The flank angles are below 0.38, while the exterior dihedral angle exceeds π/2.Source: §5.3 · p. 9
6. The remaining circumradii: a fully analytic radial estimate
The final circumradius range is handled analytically using radial transport, a tangent majorant, and contact and antipodal profiles. Monotonicity reduces the intermediate range to an endpoint calculation, while three circumradius regimes complete the proof.
6.1. Circumradius at most 0.604.
- For R ≤ R1, discarding the nonnegative G term in (6.4) gives the stated exact bound.Source: §6.1 · p. 11
6.2. Closed-form contact and antipodal profiles.
- Contact balance makes the four contact caps and four antipodal caps pairwise disjoint, and the selected profiles reach their respective levels before the disjointness threshold.Source: §6.2 · pp. 12–13
6.3. The baseline plus the antipodal cap.
- The tangent baseline plus four antipodal caps yields a contribution C(R) that strictly decreases on [R1, R0], so its minimum there is at R0.Source: §6.3 · p. 13, §6.5 · p. 16
6.4. An analytic lower bound for the positive contact cap.
- For each fixed z in [0, c], the positive-contact lower-density function ΦR(z) is nonincreasing on [R1, R0].Source: §6.4 · p. 14
6.5. Endpoint arithmetic and completion.
- 0.412753180790651 is a strict lower bound for B(D0) at the near-Jung endpoint.Source: §6.5 · p. 16
- Three circumradius ranges cover the full Jung interval, and scaling from width one to w proves theorem 1.1.Source: §6.5 · p. 17