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Symmetric Volume Descent from U4 and a Four-Dimensional Lower Bound

A mix of human-written and AI-generated textHuman understanding: all partsmath.MG — Metric Geometrymath.FA — Functional Analysismath.OC — Optimization and Control

Contributed by Hangang Wang ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We explain a symmetric deformation that decreases the volume of the four-dimensional constant-width body U4U_4. The emphasis is on why the deformation works, rather than the size of the decrease. Oddness preserves width; convexity requires control of the curvature radii. A positive cap on one support branch is coupled to a negative correction on the adjacent branch. Their first-order volume contributions have different orders in the localization parameter, so the favorable contribution dominates. A small motion towards the ball makes the direction feasible for positive time. In a separate part we give a spectral lower bound for all four-dimensional bodies of constant width one, including an explicit positive refinement. A short projection argument using a three-dimensional theorem of Jiang gives the stronger lower bound 0.178722030.17872203. A substantial gap between the lower and upper bounds remains.

Provenance statement

This research was conducted primarily during a Mitacs Globalink Research Internship at the University of Manitoba under the supervision of Professor Andriy Prymak. The initial idea of a volume-decreasing deformation of $U_4$ was developed by the author and communicated to Professor Prymak in August 2026. Professor Prymak subsequently contributed mathematical details and expanded arguments, including a higher-dimensional extension discussed in his unpublished manuscript. The manuscript combines human-written material with AI-assisted drafting, revision, and editorial refinement. AI tools also assisted with selected mathematical arguments, auxiliary checks, and code development. The author has checked and understood all parts of the manuscript, including its mathematical arguments and supporting calculations, and takes responsibility for its content and conclusions. This does not constitute formal verification or independent peer review. Supporting computational materials are available at: https://github.com/WangHangang/4d-constant-width-certificates

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References

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  4. A. Prymak, An Endpoint Variational Criterion and a Branch-Coupled Constant-Width Descent from UnU_n, n≥4n\ge4 , unpublished manuscript, 2026.
  5. H. Wang, Volume Computation and Comparison for Four-Dimensional Bodies of Constant Width , Hexagon preprint hexagon:2610.00561v1 (https://hexagonmath.org/2610.00561v1), version 1, October 26, 2026.hexagon
  6. Y. Jiang, A 0.412750.41275 lower bound for the three-dimensional Blaschke--Lebesgue problem , preprint, arXiv:2610.13964v1, 2026, https://arxiv.org/abs/2610.13964v1.arXiv
  7. D. Hug and R. Schneider, Vectorial analogues of Cauchy's surface area formula , Arch. Math. 122 (2024), 343--352, doi:10.1007/s00013-023-01962-y (https://doi.org/10.1007/s00013-023-01962-y).DOI

Version history

  1. v1Submitted by Hangang WangInitial depositCurrentOct 29, 2026