An Improved Lower Bound for the Three-Dimensional Blaschke--Lebesgue Problem from Spectral and Dual Perspectives

The Blaschke–Lebesgue problem asks for convex bodies of minimum volume among all convex bodies of prescribed constant width. In the plane, the minimizer is the Reuleaux triangle, whereas the corresponding three-dimensional problem remains open and is also known as Meissner’s conjecture. In this paper, we establish the lower bound $(4\pi \mathrm{/}33)d^3\simeq 0.380799d^3$ for the volume of any three-dimensional convex body of constant width d. This improves upon Chakerian’s lower bound, approximately $0.364916d^3$, although it remains below the volume of the conjectured minimizers, Meissner’s tetrahedra, whose volume is approximately $0.419860d^3$. The proof is based on a support-function formulation, spectral estimates via spherical harmonics, and Bochner’s formula. We also show that the resulting lower bound can be interpreted as a Lagrange dual bound for the associated concave quadratic minimization problem. This dual viewpoint suggests possible routes toward sharper lower bounds.