A 0.41275 lower bound for the three-dimensional Blaschke-Lebesgue problem

Abstract

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 $0.411040473721188\dots w^3$ 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 $\mathrm{Vol}(K)>0.41275318$ 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 $\mathrm{Vol}(K)>0.41275168$. For smaller circumradius the tangent-majorant inequality with its nonnegative gap discarded already gives $\mathrm{Vol}(K)>0.41308109$. The proof is entirely analytic and requires no computer-assisted certificate.