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 U4, the canonical raised Meissner pair, the canonical raise of U3, the body B5, and M4. 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 U4 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.
Primarily human-written textHuman understanding: all partsBodies of constant widthcertified comparisonfour-dimensional volumeinterval arithmetic
For every complex 1-unconditional convex body K⊂Cn, we prove
\[
\operatorname{ill}(K)\le
(2^{m+1}-1)\binom nm+\sum_{k=m+1}^{n}\binom nk-(n-m),
\qquad m=\lceil2n/3\rceil.
\]
This bound is asymptotic to 3n+1/πn. We further obtain
ill∗(K)≤b∗n and ill(K)=O(nlog(n+1)b∗n),
where b∗=2.4805534697… is defined by an explicit scalar equation.
For normalized bodies other than Dn that contain a full
coordinate (n−1)-polydisc, we prove the sharp bound
ill(K)≤2n+1−2. We also prove the sharp bound 14 for every
normalized complex 1-unconditional body in C3 other than
D3, and the sharp bound 30 in C4 for every
normalized non-polydisc. The four-dimensional proof includes one
finite covering assertion verified by exact integer arithmetic;
its complete verifier is included. Under full coordinate permutation
symmetry, we improve the three-dimensional bound to 12 by an explicit
twelve-vector construction and an exact finite covering certificate.
For complex Hanner bodies, we compute
both illumination numbers exactly and determine the equality cases in their dimension
bounds. If every extreme point has all coordinates nonzero, we prove
ill(K)≤2n+1−1 and a quantitative strict fractional bound
ill∗(K)<2n for normalized non-polydiscs. For real 1-symmetric
non-cubes we obtain bounds 12 in dimension four, 22 in dimension
five, and 5⋅2n−3 for every n≥6. We also determine
the leading asymptotic maximum in the real 1-symmetric non-cube
class: it is (1+o(1))2n−1. An explicit binary construction
proves this without an external coding theorem. A connection to
connected domination in cube graphs improves the upper estimate to
2n−1(1+(1+o(1))/n).
Primarily AI-generated textHuman understanding: some partsconvex geometry
We optimize a single term in the cube–Euclidean-ball construction of V. Patankar to obtain C_70 >= exp(1/(2e)) > 1.20194336847. The strongest public proposal located and reproduced in our dated source audit is Patankar's 1.201927, whose specified witness evaluates near 1.2019271425725. A separate finite-dimensional witness certifies 1.20194336. Two separately written arithmetic implementations, using Arb and rational logarithm/Stirling bounds respectively, verify the numerical comparisons and reject corrupted or excessive claims. The geometric reduction and passage to the limit are proved here. This unrefereed preprint is not peer reviewed and not formally verified; exhaustive priority is not asserted.
Primarily AI-generated textHuman understanding: some partscentrally symmetric convex bodiescube and Euclidean ballexact certificatesintrinsic volumesreverse Brunn–Minkowski inequalityrigorous arithmetic