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Volume Computation and Comparison for Four-Dimensional Bodies of Constant Width

Primarily human-written textHuman understanding: all partsmath.MG — Metric Geometrymath.NA — Numerical Analysis

Contributed by Hangang Wang ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

We compute and compare the volumes of explicit four-dimensional bodies of constant width one. The original five-entry comparison consists of the simplex-symmetric projection body U4U_4, the canonical raised Meissner pair, the canonical raise of U3U_3, the body B5B_5, and M4M_4. We combine exact geometric formulas, numerical integration, interval enclosures and inscribed rational polytopes to obtain a certified strict ordering of their volumes. We also compare U4U_4 with the direct four-dimensional AMO peabody. The two raised Meissner bodies are proved noncongruent but exactly equal in volume. An introductory account explains why different constructions benefit from different computational methods, and distinguishes approximate volume estimates from inequalities certified by exact or outward-rounded arithmetic.

Provenance statement

This work was carried out primarily during a Mitacs Globalink Research Internship at the University of Manitoba under the supervision of Professor Andriy Prymak. The author conducted the mathematical investigations and computational work and primarily wrote the initial manuscript. AI tools assisted with the organization, revision, and editorial refinement of the final manuscript, as well as selected mathematical arguments, auxiliary checks, and code development. The author has reviewed, checked, and understood the complete mathematical content of the paper and takes responsibility for its results and conclusions. The computational certificates, supporting materials, and the scope of reproducibility are documented in the manuscript and at: https://github.com/WangHangang/4d-constant-width-certificates This statement does not imply that every computational production run was independently repeated, nor does it constitute formal verification or independent peer review.

Tools used

OpenAI
ChatGPTVersion GPT-6 Sol
Anthropic
Claude OpusVersion Opus 5
OpenAI
CodexVersion GPT-6 Astra
Python Software Foundation
Python
FLINT project
python-flint (Arb)Version 0.9.0

References

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Version history

  1. v1Submitted by Hangang WangInitial depositCurrentOct 26, 2026