A 0.41275 lower bound for the three-dimensional Blaschke-Lebesgue problem
Abstract
We prove that every convex body K⊂ℝ3 of constant width w>0 satisfies Vol(K)≥ 1651/4000w3=0.41275 w3. This improves the certified lower bound 0.411040473721188… w3 announced by Hyra and the earlier spectral bound 4πw3/33 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 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 Vol(K)>0.41275168. For smaller circumradius the tangent-majorant inequality with its nonnegative gap discarded already gives Vol(K)>0.41308109. The proof is entirely analytic and requires no computer-assisted certificate.
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BibTeX
@misc{jiang2026041275,
title = {{A 0.41275 lower bound for the three-dimensional Blaschke-Lebesgue problem}},
author = {Yunjiang Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13964},
eprint = {2610.13964},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13964},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A 0.41275 lower bound for the three-dimensional Blaschke-Lebesgue problem
AU - Yunjiang Jiang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.13964
UR - https://arxiv.org/abs/2610.13964
AB - We prove that every convex body K⊂ℝ^3 of constant width w>0 satisfies Vol(K)≥ 1651/4000w^3=0.41275 w^3. This improves the certified lower bound 0.411040473721188… w^3 announced by Hyra and the earlier spectral bound 4πw^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 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 Vol(K)>0.41275168. For smaller circumradius the tangent-majorant inequality with its nonnegative gap discarded already gives Vol(K)>0.41308109. The proof is entirely analytic and requires no computer-assisted certificate.
ER -