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

  • Akatsuki Nishioka
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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)d3 ≃ 0.380799 d3 for the volume of any three-dimensional convex body of constant width d. This improves upon Chakerian's lower bound, approximately 0.364916 d3, although it remains below the volume of the conjectured minimizers, Meissner's tetrahedra, whose volume is approximately 0.419860 d3. 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.

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Cite

BibTeXRIS
BibTeX
@misc{nishioka2026improved,
  title = {{An Improved Lower Bound for the Three-Dimensional Blaschke--Lebesgue Problem from Spectral and Dual Perspectives}},
  author = {Akatsuki Nishioka},
  howpublished = {arXiv},
  year = {2026},
  month = jun,
  doi = {10.48550/arxiv.2606.01754},
  eprint = {2606.01754},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.01754},
}
RIS
TY  - UNPB
TI  - An Improved Lower Bound for the Three-Dimensional Blaschke--Lebesgue Problem from Spectral and Dual Perspectives
AU  - Akatsuki Nishioka
PY  - 2026
DA  - 2026/06/01
PB  - arXiv
DO  - 10.48550/arxiv.2606.01754
UR  - https://arxiv.org/abs/2606.01754
AB  - 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 ≃ 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.
ER  -