{"ID":2921151,"CreatedAt":"2026-06-02T02:42:49.606572591Z","UpdatedAt":"2026-06-04T05:47:54.429167893Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.01754","arxiv_id":"2606.01754","title":"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 $(4π/33)d^3 \\simeq 0.380799\\,d^3$ 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.","short_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...","url_abs":"https://arxiv.org/abs/2606.01754","url_pdf":"https://arxiv.org/pdf/2606.01754v1","authors":"[\"Akatsuki Nishioka\"]","published":"2026-06-01T06:17:20Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.MG\"]","methods":"[]","has_code":false}