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

Abstract

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 $(4\pi/33)d^3 \simeq 0.380799\,d^3$ for the volume of any three-dimensional convex body of constant width d. This improves upon Chakerian's lower bound, approximately $0.364916\,d^3$, although it remains below the volume of the conjectured minimizers, Meissner's tetrahedra, whose volume is approximately $0.419860\,d^3$. 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.