math.MG — Metric Geometry

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

Contributed by Hangang Wang

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 $U_4$, the canonical raised Meissner pair, the canonical raise of $U_3$, the body $B_5$, and $M_4$. 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 $U_4$ 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.